Multi-Antenna Jammer-Assisted Secure Short Packet Communications in IoT Networks

Abstract

In this work, we exploit a multi-antenna cooperative jammer to enable secure short packet communications in Internet of Things (IoT) networks. Specifically, we propose three jamming schemes to combat eavesdropping, i.e., the zero forcing beamforming (ZFB) scheme, null-space artificial noise (NAN) scheme, and transmit antenna selection (TAS) scheme. Assuming Rayleigh fading, we derive new closed-form approximations for the secrecy throughput with finite blocklength coding. To gain further insights, we also analyze the asymptotic performance of the secrecy throughput in the case of infinite blocklength. Furthermore, we investigate the optimization problem in terms of maximizing the secrecy throughput with the latency and reliability constraints to determine the optimal blocklength. Simulation results validate the accuracy of the approximations and evaluate the impact of key parameters such as the jamming power and the number of antennas at the jammer on the secrecy throughput.

1. Introduction

The Internet of Things (IoT), which is expected to connect various surrounding physical nodes such as different kinds of controllers and sensors, serves as a crucial architecture for e-health, home automation, smart cities, environmental monitoring, intelligent transportation, etc. [1,2,3]. Note that enormous amounts of confidential and sensitive information, e.g., financial files, trade secrets, and personal privacy, are exchanged via wireless channels in the IoT networks. For example, e-health contains personal confidential information such as physiological information and real-time location. If this private information is obtained by illegal eavesdroppers, it will result in very serious consequences. Therefore, privacy and security protection are a fundamental requirement in the design of IoT networks [4,5]. Conventionally, security for IoT is addressed by cryptographic techniques. The encryption-based security techniques have inherent high complexity in secret key generation, distribution, and management, which makes it difficult to apply for IoT networks with many resource-constrained sensors and actuators [6,7,8].

Compared with cryptographic techniques, physical layer security utilizes the inherent randomness of the wireless medium, such as fading, noise, and interference, to guarantee secure information transmission. Since physical layer security does not need the complicated key exchange procedure and provides a low-complexity and effective security solution, it is more favorable for IoT networks [9,10,11,12]. The physical layer security performance can be further enhanced by a jamming strategy. By designing an artificial noise signal, the difference between the quality of the legitimate link and the eavesdropping link can be enlarged, which will lead to a high secrecy rate. In particular, a secure downlink IoT network was considered in [13], where the power allocation between the legitimate signal and the artificial noise signal and the number of transmit antennas were jointly optimized to maximize the network secrecy throughput. A cooperative jammer in [14] sends jamming signals to confuse the eavesdroppers for downlink transmission in IoT networks. The artificial noise sent by the IoT devices was proposed in [15] for secrecy performance enhancement and the sum secrecy rate was maximized, subject to the transmit power constraint. The authors in [16] derived the secrecy outage probability of multihop IoT networks with a cooperative jamming scheme.

The above studies on IoT networks in the context of physical layer security assumed that the coding blocklength is large enough to achieve the secrecy capacity. However, the primary feature of IoT networks is the use of short packet transmissions to reduce communication latency. From [2,5,17,18], we know that the packet length in IoT applications is only a few hundreds bits, e.g., 10–300 bytes of factory automation. In this case, perfect decoding cannot be achieved even when the data rate is below the Shannon capacity. Therefore, the physical layer security based on classic infinite blocklength design cannot be directly adopted to the IoT networks with short packet transmissions. The authors in [19] evaluated the maximal achievable secrecy rate for general wiretap channels under a given information leakage, error probability, and blocklength. Based on this achievable secrecy rate, Reference [20] derived the secrecy throughput of an IoT network in the presence of a multi-antenna eavesdropper. Later, the work in [20] was extended to a wiretap channel with multiple eavesdroppers [21], multiuser multiple-input-multiple-output (MIMO) networks [22], and cognitive IoT networks with a multi-antenna relay [23]. Additionally, Reference [24] considered the relationship between secrecy throughput and secrecy coding under finite blocklength, and provided the optimal code rates and blocklength in terms of maximizing the secrecy throughput. Furthermore, the work [25] proposed some approaches to minimize the total transmit power or maximize the weighted throughput. In addition, the non-orthogonal multiple access (NOMA) technique has been incorporated to design the secure short packet communications schemes [26,27,28]. However, all these works only considered the secure communications of legitimate parties without an external helper, and most of them neglected the impact of cooperative jamming on the system performance, which may lead to an underestimation of the secrecy performance.

Motivated by the above background, in this paper, we investigate secure short packet communications in an IoT network, where an access point (AP) sends confidential information to a connected actuator in the presence of a cooperative jammer equipped with multiple antennas and an eavesdropper. Specifically, we propose three jamming schemes, i.e., the zero-forcing beamforming (ZFB) scheme, null-space artificial noise (NAN) scheme, and transmit antenna selection (TAS) scheme, for security improvement. The main contributions of our work are summarized as follows.

1.

We first derive new closed-form approximations for the secrecy throughput of an IoT network with three different jamming transmission schemes, i.e., the ZFB scheme, NAN scheme, and TAS scheme. To achieve further insights, we also investigate the asymptotic secrecy throughput in the case of infinite blocklength.

2.

We present the optimal design for the three different jamming transmission schemes. The blocklength is optimally determined in terms of maximizing the secrecy throughput subject to the constraints on the latency and reliability.

3.

The results demonstrate that both the ZFB scheme and the NAN scheme strictly outperform the TAS scheme in terms of secrecy throughput. Moreover, increasing the number of antennas at the jammer provides significant performance gains for the three proposed schemes, which can be used to compensate for the performance loss from short packet transmissions.

The rest of the paper is organized as follows. We describe the system model of an IoT network with a multi-antenna cooperative jammer in Section 2. In Section 3, we present the analytical expressions of the secrecy throughput and asymptotic secrecy throughput and characterize the maximization of the secrecy throughput. Finally, we give simulation results in Section 4 and conclude this work in Section 5.

2. System Model

We consider an IoT downlink short packet communications system, as shown in Figure 1, which consists of an AP (Alice), a legitimate actuator (Bob), a cooperative jammer, and an eavesdropper (Eve). Similar to [29,30], we assume that all nodes are equipped with a single antenna, except that the jammer has $N_j$ antennas. We also assume that all the involved channels follow quasi-static Rayleigh block fading, such that the channel coefficients keep constant within each transmission block and change independently between different blocks.

To exploit the advantages of both multiple antenna and cooperative jamming techniques, we consider three different secure transmission schemes, i.e., the ZFB scheme, NAN scheme, and TAS scheme. For the ZFB scheme, the jammer aims to maximize the interference at Eve, while avoiding interference to Bob. Thus, the beamforming vector ${\mathbf{w}}_{\mathbf{ZF}}$ is the solution of the following optimization problem:

$$\begin{array}{c}\underset{{\mathbf{w}}_{\mathbf{ZF}}}{max}\left|{\mathbf{h}}_{je}^{†}{\mathbf{w}}_{\mathbf{ZF}}\right|\hfill \\ s.t.\left|{\mathbf{h}}_{jb}^{†}{\mathbf{w}}_{\mathbf{ZF}}\right|=0\&{∥{\mathbf{w}}_{\mathbf{ZF}}∥}_F=1,\hfill \end{array}$$

(1)

where † is the conjugate transpose operator, ${∥·∥}_F$ is the Frobenius norm, ${\mathbf{h}}_{je}$ is the $N_j\times 1$ channel vector for the jammer to Eve link with entries following identical and independently distributed (i.i.d.) Rayleigh fading with parameter ${\overline{\gamma }}_{je}$, and ${\mathbf{h}}_{jb}$ is the $N_j\times 1$ channel vector for the jammer to Bob link with entries following i.i.d. Rayleigh fading with parameter ${\overline{\gamma }}_{jb}$. By using projection matrix theory [31], the beamforming vector ${\mathbf{w}}_{\mathbf{ZF}}$ can be expressed as:

$${\mathbf{w}}_{\mathbf{ZF}}=\frac{{\aleph }^{\perp }{\mathbf{h}}_{je}}{{∥{\aleph }^{\perp }{\mathbf{h}}_{je}∥}_F},$$

(2)

where ${\aleph }^{\perp }=\mathbf{I}-{\mathbf{h}}_{jb}{\left({\mathbf{h}}_{jb}^{†}{\mathbf{h}}_{jb}\right)}^{-1}{\mathbf{h}}_{jb}^{†}$ is the projection idempotent matrix with rank $N_j-1$. Accordingly, the received signal-to-noise ratios (SNRs) at Bob and Eve of the ZFB scheme, respectively, can be expressed as:

$${\gamma }_b^{ZBF}=\frac{P_a}{{\sigma }^2}{\left|h_{ab}\right|}^2$$

(3)

and

$${\gamma }_e^{ZBF}=\frac{P_a{\left|h_{ae}\right|}^2}{P_j{\left|{\mathbf{h}}_{je}^{†}{\mathbf{w}}_{\mathbf{ZF}}\right|}^2+{\sigma }^2},$$

(4)

where $P_a$ is the transmit power of Alice, $P_j$ is the transmit power of the jammer, $h_{ab}$ is the Rayleigh channel coefficient for Alice to Bob link with parameter ${\overline{\gamma }}_{ab}$, $h_{ae}$ is the Rayleigh channel coefficient for Alice to Eve link with parameter ${\overline{\gamma }}_{ae}$, and ${\sigma }^2$ is the noise variance at each receiver.

For the NAN scheme, the jammer sends artificial noise in the null space of the legitimate channel to guarantee secure communication. Thus, the beamforming matrix at the jammer is designed as $\mathbf{W}=\left[{\mathbf{w}}_{jb},{\mathbf{W}}_{je}\right]$, where ${\mathbf{w}}_{jb}\in {\mathbb{C}}^{N_j\times 1}$ is a weighted vector used for the jammer to Bob link, and ${\mathbf{W}}_{je}\in {\mathbb{C}}^{N_j\times N_j-1}$ is a weighted matrix for the null space of ${\mathbf{h}}_{jb}$, i.e., ${\mathbf{h}}_{jb}{\mathbf{W}}_{je}=0$. As in [29,32], we assume that the transmit power $P_j$ of the jammer is uniformly allocated in the $N_j-1$ directions. Accordingly, the received SNRs at Bob and Eve of the NAN scheme, respectively, can be expressed as:

$${\gamma }_b^{NAN}=\frac{P_a}{{\sigma }^2}{\left|h_{ab}\right|}^2$$

(5)

and

$${\gamma }_e^{NAN}=\frac{P_a{\left|h_{ae}\right|}^2}{\frac{P_j}{N_j-1}{∥{\mathbf{h}}_{je}^{†}{\mathbf{W}}_{je}∥}_F^2+{\sigma }^2}.$$

(6)

In addition to transmit beamforming, TAS is regarded as another effective method to improve the physical layer security performance. Moreover, TAS is particularly well adapted to systems with computational constraints, since it has two main advantages: low cost and easy implementation. For the TAS scheme, the jammer selects an antenna, which minimizes the interference imposed on Bob, to transmit the jamming signal. Accordingly, the received SNRs at Bob and Eve of the TAS scheme, respectively, can be expressed as:

$${\gamma }_b^{TAS}=\frac{P_a{\left|h_{ab}\right|}^2}{P_j\underset{1\le i\le N_j}{min}{\left|h_{j_{i}b}\right|}^2+{\sigma }^2}$$

(7)

and

$${\gamma }_e^{TAS}=\frac{P_a{\left|h_{ae}\right|}^2}{P_j{\left|h_{j_{i^{*}}e}\right|}^2+{\sigma }^2},$$

(8)

where $h_{j_{i}b}$ is the Rayleigh channel coefficient for the i-th antenna of the jammer to Bob link, and $h_{j_{i^{*}}e}$ is the Rayleigh channel coefficient for the selected antenna of the jammer to Eve link.

According to [19], the maximal instantaneous secrecy rate of the IoT downlink short packet communications system with a cooperative jammer for a given information leakage probability $\delta$, decoding error probability $ϵ$, and blocklength N can be approximated as:

$$R_s^{★}\left(N,ϵ,\delta \right)=\left\{\begin{array}{cc}{C}_s-\sqrt{\frac{V_b^{★}}{N}}\frac{Q^{-1}\left(ϵ\right)}{ln2}-\sqrt{\frac{V_e^{★}}{N}}\frac{Q^{-1}\left(\delta \right)}{ln2},& {\gamma }_b^{★}>{\gamma }_e^{★},\\ 0,& {\gamma }_b^{★}\le {\gamma }_e^{★},\end{array}\right.$$

(9)

where $C_s={log}_2\left(1+{\gamma }_b^{★}\right)-{log}_2\left(1+{\gamma }_e^{★}\right)$ is the secrecy capacity for the infinite number of channel uses, $V_x^{★}=1-{\left(1+{\gamma }_x^{★}\right)}^{-2}$, $x\in \left\{b,e\right\}$ is the channel dispersion, $Q^{-1}\left(·\right)$ is the inverse Q-function $Q\left(x\right)={\int }_x^{\infty }\frac{1}{\sqrt{2\pi }}{e}^{-\frac{t^2}{2}}dt$, and $★\in \left\{ZFB,NAN,TAS\right\}$. For the reader’s convenience, we define ${\lambda }_a=\frac{P_a}{{\sigma }^2}$ and ${\lambda }_j=\frac{P_j}{{\sigma }^2}$.

3. Secrecy Performance Analysis

In this section, we give a comprehensive analysis on the secrecy performance of the three proposed schemes. We first derive closed-form approximations for the secrecy throughput. Then, we analyze the asymptotic secrecy throughput in the case of infinite blocklength to gain further insights. Furthermore, the optimization problem in terms of maximizing the secrecy throughput is investigated under the latency and reliability constraints.

3.1. Secrecy Throughput

The secrecy throughput is the average secrecy rate where confidential messages from Alice are reliably transmitted to Bob under a certain secrecy constraint. Mathematically, it can be expressed as:

$$T=\mathbb{E}{}_{{\gamma }_b,{\gamma }_e}\left(\frac{B}{N}\left(1-ϵ\right)\right)=\frac{B}{N}\left(1-\overline{ϵ}\right)$$

(10)

where B is the information bits delivered over N channel uses and $\overline{ϵ}=\mathbb{E}{}_{{\gamma }_b,{\gamma }_e}\left(ϵ\right)$ is the average decoding error probability.

3.1.1. ZFB Scheme

The closed-form expression for the secrecy throughput of the IoT downlink short packet communications system with the ZFB scheme can be approximated as (here, we assume that $N_j\ge 3$; when $N_j\le 3$, the result can be directly derived by similar procedures):

$$\begin{array}{cc}\hfill T^{ZFB}& \approx \frac{M_{1}B}{2N}\sum _{m=1}^{M_2}\left(\frac{\pi }{M_2}\sqrt{1-t_m^2}{e}^{-\frac{{\varpi }_m}{{\lambda }_a{\overline{\gamma }}_{ab}}}\left(\frac{e^{-\frac{M_1\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\lambda }_a{\overline{\gamma }}_{ae}{\overline{\gamma }}_{je}^{N_j-1}}{\left(\frac{M_1{\lambda }_j\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}+\frac{1}{{\overline{\gamma }}_{je}}\right)}^{1-N_j}\right.\right.\hfill \\ \hfill & \left.\left.+\frac{{\lambda }_j\left(N_j-1\right)}{{\lambda }_a{\overline{\gamma }}_{ae}{\overline{\gamma }}_{je}^{N_j-1}}{e}^{-\frac{M_1\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}}{\left(\frac{M_1{\lambda }_j\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}+\frac{1}{{\overline{\gamma }}_{je}}\right)}^{-N_j}\right)\right)\hfill \\ \hfill & +\frac{B{\lambda }_a^{N_j-2}{\overline{\gamma }}_{ae}^{N_j-2}{e}^{-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}}{N{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}}\left(\sum _{k=1}^{N_j-2}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{\left(-1\right)}^{N_j-2-k}{e}^{-\frac{M_1{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{M_1}{{\lambda }_a{\overline{\gamma }}_{ae}}}}{\left(N_j-2\right)!{\left(M_1{\lambda }_j{\overline{\gamma }}_{je}+{\lambda }_a{\overline{\gamma }}_{ae}\right)}^k}\right.\hfill \\ \hfill & \times {\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2-k}-\frac{{\left(-1\right)}^{N_j-2}}{\left(N_j-2\right)!}{\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2}\hfill \\ \hfill & \left.\times e^{\frac{{\varpi }_1{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}}Ei\left(-\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)\left(M_1+\frac{{\lambda }_a{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\right)+e^{-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}\hfill \\ \hfill & \times \frac{B{\lambda }_a^{N_j-1}{\overline{\gamma }}_{ae}^{N_j-1}\left(N_j-1\right)e^{-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}}{N{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}\left(N_j-1\right)!}\left(\sum _{k=1}^{N_j-1}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{e}^{-\frac{M_1{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{M_1}{{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\left(M_1{\lambda }_j{\overline{\gamma }}_{je}+{\lambda }_a{\overline{\gamma }}_{ae}\right)}^k}\right.\hfill \\ \hfill & \times {\left(-\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1-k}-{\left(-\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1}\hfill \\ \hfill & \times e^{\frac{{\varpi }_1{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}}\left.Ei\left(-\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)\left(M_1+\frac{{\lambda }_a{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\right),\hfill \end{array}$$

(11)

where $Ei\left(·\right)$ is the exponential integral function and $M_1$, $M_2$, ${\varpi }_1$, ${\varpi }_m$, and $t_m$ are defined in Appendix A.

Proof. 

See Appendix A. ☐

3.1.2. NAN Scheme

The closed-form expression for the secrecy throughput of the IoT downlink short packet communications system with the NAN scheme can be approximated as (here, we assume that $N_j\ge 3$; when $N_j\le 3$, the result can be directly derived by similar procedures):

$$\begin{array}{cc}\hfill T^{NAN}& \approx \frac{M_{1}B}{2N}\sum _{m=1}^{M_2}\left(\frac{\pi }{M_2}\sqrt{1-t_m^2}{e}^{-\frac{{\varpi }_m}{{\lambda }_a{\overline{\gamma }}_{ab}}}\left({\left(\frac{M_1{\lambda }_j\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}\left(N_j-1\right)}+\frac{1}{{\overline{\gamma }}_{je}}\right)}^{1-N_j}\right.\right.\hfill \\ \hfill & \left.\left.\times \frac{e^{-\frac{M_1\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\lambda }_a{\overline{\gamma }}_{ae}{\overline{\gamma }}_{je}^{N_j-1}}+\frac{{\lambda }_j{e}^{-\frac{M_1\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\lambda }_a{\overline{\gamma }}_{ae}{\overline{\gamma }}_{je}^{N_j-1}}{\left(\frac{M_1{\lambda }_j\left(t_m+1\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}\left(N_j-1\right)}+\frac{1}{{\overline{\gamma }}_{je}}\right)}^{-N_j}\right)\right)\hfill \\ \hfill & +\frac{B{\lambda }_a^{N_j-2}{\overline{\gamma }}_{ae}^{N_j-2}{e}^{-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}}{N{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}}\left(\sum _{k=1}^{N_j-2}\frac{{\left(N_j-1\right)}^{N_j-1}\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{e}^{-\frac{M_1{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{M_1}{{\lambda }_a{\overline{\gamma }}_{ae}}}}{\left(N_j-2\right)!{\left(M_1{\lambda }_j{\overline{\gamma }}_{je}+{\lambda }_a{\overline{\gamma }}_{ae}\left(N_j-1\right)\right)}^k}\right.\hfill \\ \hfill & \times {\left(-\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2-k}-\frac{e^{\frac{{\varpi }_1{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{N_j-1}{{\lambda }_j{\overline{\gamma }}_{je}}}}{\left(N_j-2\right)!}{\left(-\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2}\hfill \\ \hfill & \times \left.{\left(N_j-1\right)}^{N_j-1}Ei\left(-\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)\left(M_1+\frac{{\lambda }_a{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\right)\hfill \\ \hfill & +\frac{B{\lambda }_a^{N_j-1}{\overline{\gamma }}_{ae}^{N_j-1}{\left(N_j-1\right)}^{N_j}{e}^{-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}}{N{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}\left(N_j-1\right)!}\left(\sum _{k=1}^{N_j-1}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{\left(-1\right)}^{N_j-1-k}{e}^{-\frac{M_1}{{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\left(M_1{\lambda }_j{\overline{\gamma }}_{je}+{\lambda }_a{\overline{\gamma }}_{ae}\left(N_j-1\right)\right)}^k}\right.\hfill \\ \hfill & \times {\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1-k}{e}^{-\frac{M_1{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}}-{\left(-\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1}{e}^{\frac{N_j-1}{{\lambda }_j{\overline{\gamma }}_{je}}}\hfill \\ \hfill & \times e^{\frac{{\varpi }_1{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}}\left.Ei\left(-\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)\left(M_1+\frac{{\lambda }_a{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\right).\hfill \end{array}$$

(12)

Proof. 

Since ${∥{\mathbf{h}}_{je}^{†}{\mathbf{W}}_{je}∥}_F^2$ is a chi-squared random variable with $2\left(N_j-1\right)$ degrees of freedom [33], by following similar steps as the ZFB scheme, we can obtain the desired expression in (12) after some mathematical manipulations. □

3.1.3. TAS Scheme

The closed-form expression for the secrecy throughput of the IoT downlink short packet communications system with the TAS scheme can be approximated as (here, we assume that $c_1\ne \frac{{\lambda }_a{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}}$; when $c_1=\frac{{\lambda }_a{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}}$, the result can be obtained by similar analysis):

$$\begin{array}{cc}\hfill T^{TAS}& \approx \frac{M_{1}B}{2N}\sum _{m=1}^{M_2}\left(\frac{\pi N_j{\lambda }_a{\overline{\gamma }}_{ab}\sqrt{1-t_m^2}{e}^{-\frac{{\varpi }_m}{{\lambda }_a{\overline{\gamma }}_{ab}}}}{M_2{\varpi }_m{\lambda }_j{\overline{\gamma }}_{jb}+M_2{\lambda }_a{\overline{\gamma }}_{ab}{N}_j}\left(\frac{2e^{-\frac{M_1\left(t_{m+1}\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\lambda }_j{\overline{\gamma }}_{je}{M}_1\left(t_{m+1}\right)+2{\lambda }_a{\overline{\gamma }}_{ae}}\right.\right.\hfill \\ \hfill & \left.\left.+\frac{4{\lambda }_a{\overline{\gamma }}_{ae}{\lambda }_j{\overline{\gamma }}_{je}{e}^{-\frac{M_1\left(t_{m+1}\right)}{2{\lambda }_a{\overline{\gamma }}_{ae}}}}{{\left({\lambda }_j{\overline{\gamma }}_{je}{M}_1\left(t_{m+1}\right)+2{\lambda }_a{\overline{\gamma }}_{ae}\right)}^2}\right)\right)-\frac{B{N}_j{\lambda }_a^2{\overline{\gamma }}_{ab}{\overline{\gamma }}_{ae}{\overline{\gamma }}_{je}{e}^{-\frac{M_1{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{M_1}{{\lambda }_a{\overline{\gamma }}_{ae}}-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}}{N{\varpi }_1{\overline{\gamma }}_{jb}\left(M_1{\lambda }_j{\overline{\gamma }}_{je}+{\lambda }_a{\overline{\gamma }}_{ae}\right)}\hfill \\ \hfill & \times \frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}-c_1{\lambda }_j{\overline{\gamma }}_{je}}+\left(\frac{{\lambda }_a{\lambda }_j{\overline{\gamma }}_{je}}{{\left({\lambda }_a{\overline{\gamma }}_{ae}-c_1{\lambda }_j{\overline{\gamma }}_{je}\right)}^2}-\frac{{\varpi }_1}{\left({\lambda }_a{\overline{\gamma }}_{ae}-c_1{\lambda }_j{\overline{\gamma }}_{je}\right){\overline{\gamma }}_{ab}}\right)\hfill \\ \hfill & \times \frac{B{N}_j{\lambda }_a{\overline{\gamma }}_{ab}{\overline{\gamma }}_{ae}}{N{\varpi }_1{\lambda }_j{\overline{\gamma }}_{jb}}{e}^{\frac{{\varpi }_1{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}}Ei\left(-\left(\frac{{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)\left(M_1+\frac{{\lambda }_a{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\hfill \\ \hfill & \times e^{-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}-\frac{B{N}_j{\lambda }_a{\overline{\gamma }}_{ab}{\overline{\gamma }}_{ae}}{N{\varpi }_1{\lambda }_j{\overline{\gamma }}_{jb}}\left(\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}-c_1{\lambda }_j{\overline{\gamma }}_{je}}+\frac{{\lambda }_a{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ae}}{{\left({\lambda }_a{\overline{\gamma }}_{ae}-c_1{\lambda }_j{\overline{\gamma }}_{je}\right)}^2}\right)\hfill \\ \hfill & \times e^{\frac{c_1{\varpi }_1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{c_1}{{\lambda }_a{\overline{\gamma }}_{ae}}-\frac{{\varpi }_1-1}{{\lambda }_a{\overline{\gamma }}_{ab}}}Ei\left(-\frac{{\varpi }_1\left(M_1+c_1\right)}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{M_1+c_1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right),\hfill \end{array}$$

(13)

where $c_1=\frac{\left({\varpi }_1-1\right){\lambda }_j{\overline{\gamma }}_{jb}+{\lambda }_a{\overline{\gamma }}_{ab}{N}_j}{{\varpi }_1{\lambda }_j{\overline{\gamma }}_{jb}}$.

Proof. 

Since ${\left|h_{j_{i^{*}}e}\right|}^2$ is an exponentially distributed random variable and the probability density function (PDF) of $Z_1=\underset{1\le i\le N_j}{min}{\left|h_{j_{i}b}\right|}^2$ is given by $f_{Z_1}\left(z\right)=\frac{N_j}{{\overline{\gamma }}_{jb}}{e}^{-\frac{N_{j}z}{{\overline{\gamma }}_{jb}}}$ [29], by following a similar procedure as the ZFB scheme, we can derive the desired result in (13) after some mathematical manipulations.

The derived analytical results in (11)–(13) provide an efficient means to evaluate the secrecy throughput of the IoT downlink short packet communications system with the three proposed transmission schemes. Moreover, the approximations for the secrecy throughput of finite blocklength will be verified via simulations. □

3.2. Asymptotic Analysis

In an effort to understand the relationship between infinite blocklength and finite blocklength and gain more physical insights, in this subsection, we look into the asymptotic secrecy throughput under the infinite blocklength regime, i.e., $N\to \infty$. When $N\to \infty$, we have $ϵ\to 0$ as long as ${\gamma }_b^{★}>{\gamma }_e^{★}$ from (9). Therefore, the asymptotic secrecy throughput in the case of infinite blocklength can be mathematically expressed as:

$$T_{N\to \infty }^{★}=\frac{B}{N}Pr\left({\gamma }_b^{★}>{\gamma }_e^{★}\right).$$

(14)

Corollary 1. 

The asymptotic secrecy throughput of the ZFB scheme in the infinite blocklength regime is given by:

$$\begin{array}{cc}\hfill T_{N\to \infty }^{ZFB}& =B\left({\Delta }_1+{\Delta }_2\right)N^{-1},\hfill \end{array}$$

(15)

where ${\Delta }_1$ and ${\Delta }_2$ are given by:

$$\begin{array}{cc}\hfill {\Delta }_1& =\frac{{\lambda }_a^{N_j-2}{\overline{\gamma }}_{ae}^{N_j-2}}{{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}}\left(\sum _{k=1}^{N_j-2}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{\left(-1\right)}^{N_j-2-k}}{\left(N_j-2\right)!{\lambda }_a^k{\overline{\gamma }}_{ae}^k}{\left(\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2-k}\right.\hfill \\ \hfill & \left.-\frac{e^{\frac{{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}}}{\left(N_j-2\right)!}{\left(-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2}Ei\left(-\frac{{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill {\Delta }_2& =\frac{{\lambda }_a^{N_j-1}{\overline{\gamma }}_{ae}^{N_j-1}\left(N_j-1\right)}{{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}}\left(\sum _{k=1}^{N_j-1}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k}{\left(N_j-1\right)!{\lambda }_a^k{\overline{\gamma }}_{ae}^k}{\left(-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1-k}\right.\hfill \\ \hfill & -\frac{e^{\frac{{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}}}{\left(N_j-1\right)!}{\left(-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1}\left.Ei\left(-\frac{{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right).\hfill \end{array}$$

(17)

Corollary 2. 

The asymptotic secrecy throughput of the NAN scheme in the infinite blocklength regime is given by:

$$\begin{array}{cc}\hfill T_{N\to \infty }^{NAN}& =B\left({\Delta }_3+{\Delta }_4\right)N^{-1},\hfill \end{array}$$

(18)

where ${\Delta }_3$ and ${\Delta }_4$ are given by:

$$\begin{array}{cc}\hfill {\Delta }_3& =\frac{{\lambda }_a^{N_j-2}{\overline{\gamma }}_{ae}^{N_j-2}{\left(N_j-1\right)}^{N_j-1}}{{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}}\left(\sum _{k=1}^{N_j-2}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{\left(-1\right)}^{N_j-2-k}}{\left(N_j-2\right)!{\lambda }_a^k{\overline{\gamma }}_{ae}^k{\left(N_j-1\right)}^k}{\left(\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2-k}\right.\hfill \\ \hfill & \left.-\frac{e^{\frac{{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{N_j-1}{{\lambda }_j{\overline{\gamma }}_{je}}}}{\left(N_j-2\right)!}{\left(-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-2}Ei\left(-\frac{{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}-\frac{N_j-1}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right)\hfill \end{array}$$

(19)

and

$$\begin{array}{cc}\hfill {\Delta }_4& =\frac{{\lambda }_a^{N_j-1}{\overline{\gamma }}_{ae}^{N_j-1}{\left(N_j-1\right)}^{N_j}}{{\lambda }_j^{N_j-1}{\overline{\gamma }}_{je}^{N_j-1}}\left(\sum _{k=1}^{N_j-1}\frac{\left(k-1\right)!{\lambda }_j^k{\overline{\gamma }}_{je}^k{\left(-1\right)}^{N_j-1-k}}{\left(N_j-1\right)!{\lambda }_a^k{\overline{\gamma }}_{ae}^k{\left(N_j-1\right)}^k}\right.{\left(\frac{{\overline{\gamma }}_{ae}+{\overline{\gamma }}_{ab}}{{\lambda }_a{\overline{\gamma }}_{ab}{\overline{\gamma }}_{ae}}\right)}^{N_j-1-k}\hfill \\ \hfill & -\frac{e^{\frac{{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{N_j-1}{{\lambda }_j{\overline{\gamma }}_{je}}}}{\left(N_j-1\right)!}{\left(-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_a{\overline{\gamma }}_{ae}}\right)}^{N_j-1}\left.Ei\left(-\frac{{\overline{\gamma }}_{ae}\left(N_j-1\right)}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}-\frac{N_j-1}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\right).\hfill \end{array}$$

(20)

Corollary 3. 

The asymptotic secrecy throughput of the TAS scheme in the infinite blocklength regime is given by:

$$T_{N\to \infty }^{TAS}=B\left({\Delta }_5-{\Delta }_6-{\Delta }_7\right)N^{-1},$$

(21)

where ${\Delta }_7=\frac{N_j{\overline{\gamma }}_{ab}{\overline{\gamma }}_{je}}{{\overline{\gamma }}_{jb}{\overline{\gamma }}_{ae}-N_j{\overline{\gamma }}_{ab}{\overline{\gamma }}_{je}}$ and ${\Delta }_5$ and ${\Delta }_6$ are given by:

$$\begin{array}{cc}\hfill {\Delta }_5& =\frac{N_j{\overline{\gamma }}_{ae}\left({\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{jb}{\overline{\gamma }}_{ab}-{\overline{\gamma }}_{ae}{\overline{\gamma }}_{jb}+{\overline{\gamma }}_{ab}{\overline{\gamma }}_{je}{N}_j\right)}{{\lambda }_j{\left({\overline{\gamma }}_{ae}{\overline{\gamma }}_{jb}-{\overline{\gamma }}_{ab}{\overline{\gamma }}_{je}{N}_j\right)}^2}\hfill \\ \hfill & \times e^{\frac{{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}+\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}}Ei\left(-\frac{{\overline{\gamma }}_{ae}}{{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{ab}}-\frac{1}{{\lambda }_j{\overline{\gamma }}_{je}}\right)\hfill \end{array}$$

(22)

and

$$\begin{array}{cc}\hfill {\Delta }_6& =\frac{N_j{\overline{\gamma }}_{ab}\left({\overline{\gamma }}_{ae}{\overline{\gamma }}_{jb}-{\overline{\gamma }}_{ab}{\overline{\gamma }}_{je}{N}_j+{\lambda }_j{\overline{\gamma }}_{je}{\overline{\gamma }}_{jb}{\overline{\gamma }}_{ae}\right)}{{\lambda }_j{\left({\overline{\gamma }}_{ae}{\overline{\gamma }}_{jb}-{\overline{\gamma }}_{ab}{\overline{\gamma }}_{je}{N}_j\right)}^2}\hfill \\ \hfill & \times e^{\frac{N_j}{{\lambda }_j{\overline{\gamma }}_{jb}}+\frac{{\overline{\gamma }}_{ab}{N}_j}{{\lambda }_j{\overline{\gamma }}_{jb}{\overline{\gamma }}_{ae}}}Ei\left(-\frac{N_j}{{\lambda }_j{\overline{\gamma }}_{jb}}-\frac{{\overline{\gamma }}_{ab}{N}_j}{{\lambda }_j{\overline{\gamma }}_{jb}{\overline{\gamma }}_{ae}}\right).\hfill \end{array}$$

(23)

According to (15), (18), and (21), the secrecy throughput of the three proposed transmission schemes approaches zero as $N\to \infty$. However, the secrecy capacity of the three proposed transmission schemes is not zero for this particular case. Moreover, the asymptotic secrecy throughput of the three proposed transmission schemes is independent of the parameter $\delta$. This is because the secrecy rate $\frac{B}{N}$ can always be achieved without leaking any information to the eavesdropper in the infinite blocklength regime, when ${\gamma }_b^{★}>{\gamma }_e^{★}$.

3.3. Secrecy Throughput Maximization

We note that there exists a transmission latency–reliability tradeoff introduced by the blocklength. As such, in this subsection, we determine the optimal blocklength in terms of maximizing the secrecy throughput subject to the given latency and reliability constraints. Mathematically, the problem is formulated as:

$$\begin{array}{cc}\hfill \underset{N}{max}& T^{★},\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill \hspace{1em} s.t.& \overline{ϵ}\le {ϵ}_{max},\hfill \end{array}$$

(24b)

$$\begin{array}{cc}\hfill & N\le N_{max},\hfill \end{array}$$

(24c)

$$\begin{array}{cc}\hfill & N\in {\mathbb{N}}^{+},\hfill \end{array}$$

(24d)

where (24b) is the reliability constraint of the considered system, (24c) is the transmission latency constraint of the considered system, and ${\mathbb{N}}^{+}$ is the non-negative integer set.

Next, we will clarify that $T^{★}$ is quasi-concave with respect to N. In the first step, by taking the first-order and second-order derivative of $ϵ$ on N, we have:

$$\frac{\partial ϵ}{\partial N}=\frac{\partial ϵ}{\partial \varphi }\frac{\partial \varphi }{\partial N}$$

(25)

and

$$\frac{{\partial }^2ϵ}{\partial N^2}=\frac{{\partial }^2ϵ}{\partial {\varphi }^2}{\left(\frac{\partial \varphi }{\partial N}\right)}^2+\frac{\partial ϵ}{\partial \varphi }\frac{{\partial }^2\varphi }{\partial N^2},$$

(26)

where $\varphi =\sqrt{\frac{N}{V_b^{★}}}\left(ln\frac{1+{\gamma }_b^{★}}{1+{\gamma }_e^{★}}-\sqrt{\frac{V_e^{★}}{N}}{Q}^{-1}\left(\delta \right)-\frac{B}{N}ln2\right)$. Note that $ϵ$ is generally much smaller than $0.5$ to ensure high reliability [34]; thus, we have $\varphi =Q^{-1}\left(ϵ\right)>0$, $\frac{\partial ϵ}{\partial \varphi }=-\frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\varphi }^2}{2}}<0$, and $\frac{{\partial }^2ϵ}{\partial {\varphi }^2}=\frac{\varphi }{\sqrt{2\pi }}{e}^{-\frac{{\varphi }^2}{2}}>0$. Then, we analyze the sign of $\frac{\partial \varphi }{\partial N}$ and $\frac{{\partial }^2\varphi }{\partial N^2}$. By taking the first-order and second-order derivative of $\varphi$ on N, we have:

$$\frac{\partial \varphi }{\partial N}=\frac{\frac{N}{\sqrt{V_b^{★}}}ln\frac{1+{\gamma }_b^{★}}{1+{\gamma }_e^{★}}+\frac{Bln2}{\sqrt{V_b^{★}}}}{2N^{3/\phantom{32}2}}$$

(27)

and

$$\frac{{\partial }^2\varphi }{\partial N^2}=-\frac{\frac{N}{\sqrt{V_b^{★}}}ln\frac{1+{\gamma }_b^{★}}{1+{\gamma }_e^{★}}+\frac{3Bln2}{\sqrt{V_b^{★}}}}{4N^{5/\phantom{52}2}}.$$

(28)

From (27) and (28), it is straightforward to obtain that $\frac{\partial \varphi }{\partial N}>0$ and $\frac{{\partial }^2\varphi }{\partial N^2}<0$. Thus, we know that $ϵ$ is a convex decreasing function with respect to N. Based on the Leibniz integral rule, $\overline{ϵ}$ is also a convex decreasing function with respect to N. Accordingly, $T^{★}$ is a quasi-concave function with respect to N.

Based on the above discussions, problem (24) can be simplified as:

$$\begin{array}{cc}\hfill \underset{N}{max}& T^{★},\hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill s.t.& N^o\le N\le N_{max},\hfill \end{array}$$

(29b)

$$\begin{array}{cc}\hfill & N\in {\mathbb{N}}^{+},\hfill \end{array}$$

(29c)

where $N^o$ is the solution of $\overline{ϵ}={ϵ}_{max}$. We clarify that the optimal blocklength in terms of maximizing the secrecy throughput exists only when $⌈N^o⌉\le N_{max}$, where $⌈·⌉$ is the ceiling operation. Now, we present the optimal blocklength in terms of maximizing the secrecy throughput in the following corollary.

Corollary 4. 

When $⌈N^o⌉\le N_{max}$, the optimal blocklength in terms of maximizing the secrecy throughput is:

$$N^{*}=\left\{\begin{array}{cc}⌈N^o⌉,& N^{\#}\le N^o,\\ arg{max}_{N\in \left\{⌈N^{\#}⌉,⌊N^{\#}⌋\right\}}{T}^{*},& N^o<N^{\#}<N_{max},\\ N_{max},& N^{\#}\ge N_{max},\end{array}\right.$$

(30)

where $N^{\#}$ is the solution of $\frac{\partial T^{★}}{\partial N}=0$ and $⌊·⌋$ is the floor operation.

Proof. 

Firstly, the integer constraint in problem (29) is relaxed. Then, based on the fact that $\overline{ϵ}$ is a convex decreasing function with respect to N and $T^{★}$ is a quasi-concave function with respect to N, the optimal blocklength can be derived directly. ☐

4. Simulation Results

In this section, computer simulations by Matlab are provided to verify the analytical results of the three proposed transmission schemes and evaluate the secrecy throughput performance of the three proposed transmission schemes. In simulations, unless specified otherwise, we assume that $N_j=5$, $\delta ={10}^{-2}$, $M_1=20$, $M_2=1500$, ${\overline{\gamma }}_{ab}={\overline{\gamma }}_{ae}={\overline{\gamma }}_{jb}={\overline{\gamma }}_{je}=1$, and ${\sigma }^2=0$ dBm. From the figures, we can see that the analytical results coincide well with the computer simulation points, which demonstrates the correctness of our analysis.

Figure 2, Figure 3 and Figure 4 plot the secrecy throughput of the ZFB, NAN, and TAS schemes with different values of B, respectively. It is clear that the secrecy throughput first increases and then decreases as N increases, i.e., an optimal value for N exists to maximize the secrecy throughput. This phenomenon is explained as follows: when N is relatively small, increasing N is beneficial to improve the transmission reliability, which, of course, will lead to a higher secrecy throughput. However, when N is very large, the communication latency is large, which, consequently, will result in the degradation of the secrecy throughput. The second observation is that the optimal blocklength increases as B increases. This is because compared with communication latency, the decoding error at Bob becomes more obvious when the number of information bits gets larger.

Figure 5 investigates the impact of the transmit power $P_a$ on the secrecy throughput with the three proposed schemes. Moreover, we present a benchmark scheme for comparison, i.e., without the jammer scheme, which consists of a source, a destination, and an eavesdropper, as in [20]. It is clear that the secrecy throughput of all the proposed schemes increases first and then stays constant as $P_a$ increases. This is because the ratio of average channel gain on the legitimate link to the eavesdropping link becomes the bottleneck of the secrecy throughput in the high-SNR regime. In addition, we observe that the ZFB and NAN schemes always attain better performance than the TAS and benchmark schemes. This is because the jamming signals of the ZFB scheme and the NAN scheme only selectively interfere with the eavesdropper and have no impact on the signal reception of the legitimate user. Thus, the multi-antenna jammer can be employed to improve the system performance.

Figure 6 shows the impact of the jamming power $P_j$ on the secrecy throughput with the three proposed schemes. We first observe that when $P_j$ is sufficiently large, the secrecy throughput converges to a constant for the ZFB and NAN schemes, whereas it goes to zero for the TAS scheme. This can be explained by the fact that the use of the TAS scheme not only degrades the channel condition of the eavesdropper, which is beneficial to secure transmission, but also deteriorates the channel condition of the legitimate user, which is adverse to secure transmission. It is also worth noting that the TAS scheme outperforms the benchmark scheme in certain regimes, i.e., when the jamming power is small. Therefore, the designer has to carefully choose the jamming power of the TAS scheme.

Figure 7 examines the secrecy throughput of the three proposed schemes versus the number of antennas at the jammer. It is clear that the secrecy throughput of the three proposed schemes increases as the number of antennas at the jammer increases. Therefore, we conclude that increasing the number of antennas at the jammer can compensate for the performance loss from short packet transmissions. In particular, the TAS scheme can achieve more significant performance gains by adding more antennas at the jammer. This is attributed to the fact that increasing the number of antennas at the jammer will make it easier to select a better antenna, which reduces the jamming interference at the legitimate user.

Figure 8 depicts the optimal blocklength and the corresponding maximum secrecy throughput versus the transmit power $P_a$ subject to the latency and reliability constraints. When the optimization problem in terms of maximizing the secrecy throughput is infeasible, we set the optimal blocklength to zero and the maximum secrecy throughput is set to zero. We first observe that one critical point exists and when $P_a$ does not exceed this point, the maximum secrecy throughput is zero. This is not surprising, since when $P_a$ is small, it is unable to meet the reliability constraint. We further observe that both the optimal blocklength and the maximum secrecy throughput converge to nonzero constants as $P_a\to \infty$. This is because the secrecy throughput of the considered system is independent of $P_a$ in the high SNR regime.

Figure 9 illustrates the optimal blocklength and the corresponding maximum secrecy throughput versus the jamming power $P_j$ subject to the latency and reliability constraints. We first observe that when $P_j$ is either extremely small or large, the maximum secrecy throughput of the TAS scheme approaches zero. This is because the decoding error probability of the TAS scheme is large in the two extreme cases. Therefore, the TAS scheme can only achieve its best performance for in-between values of $P_j$. We further observe that the optimal blocklength of the TAS scheme first decreases and then increases as $P_j$ increases. This is because there exists a tradeoff between decoding error and transmission latency.

5. Conclusions

In this paper, we investigated the secrecy performance of IoT networks with finite blocklength coding. To exploit the available multi-antenna jammer for secrecy enhancement, we proposed three different secure transmission schemes. Specifically, we derived approximate closed-form expressions for the secrecy throughput of all the proposed schemes. Moreover, we presented the asymptotic secrecy throughput in the case of infinite blocklength to gain further insights. In addition, we determined the optimal blocklength that maximizes the secrecy throughput subject to the latency and reliability constraints. Numerical results demonstrated that both the ZFB scheme and the NAN scheme strictly outperform the TAS scheme in terms of the secrecy throughput, and the performance loss from short packet transmissions can be compensated for by increasing the number of antennas at the jammer.