Beta Distribution Function for Cooperative Spectrum Sensing against Byzantine Attack in Cognitive Wireless Sensor Networks

Abstract

In order to explore more spectrum resources to support sensors and their related applications, cognitive wireless sensor networks (CWSNs) have emerged to identify available channels being underutilized by the primary user (PU). To improve the detection accuracy of the PU signal, cooperative spectrum sensing (CSS) among sensor paradigms is proposed to make a global decision about the PU status for CWSNs. However, CSS is susceptible to Byzantine attacks from malicious sensor nodes due to its open nature, resulting in wastage of spectrum resources or causing harmful interference to PUs. To suppress the negative impact of Byzantine attacks, this paper proposes a beta distribution function (BDF) for CSS among multiple sensors, which includes a sequential process, beta reputation model, and weight evaluation. Based on the sequential probability ratio test (SPRT), we integrate the proposed beta reputation model into SPRT, while improving and reducing the positive and negative impacts of reliable and unreliable sensor nodes on the global decision, respectively. The numerical simulation results demonstrate that, compared to SPRT and weighted sequential probability ratio test (WSPRT), the proposed BDF has outstanding effects in terms of the error probability and average number of samples under various attack ratios and probabilities.

1. Introduction

1.1. Background

The technology related to sensors has gradually become popular and is widely used in fields such as smart homes, fire detection, animal tracking, healthcare monitoring [1], etc. Over the last decades, we have been witnesses to the rapid development of wireless sensor networks, mainly due to their high cost-effectiveness and convenience in comparison with traditional wire-based communication technologies [2]. However, limited spectrum resources are no longer sufficient to meet the widely used and growing demand for sensors and their related applications and services. According to research by the Federal Communications Commission (FCC), the spectrum resources in a network area or system are divided into multiple non-adjacent small blocks (a. k. a. spectrum hole) in both time and space domains (commercial or technical perspective). This fragmentation of spectrum division leads to a decrease in spectrum utilization and affects capacity and performance. However, in order to popularize wireless sensor networks and improve their service quality, cognitive radio (CR) empowerment technology has emerged to support sensors in identifying and utilizing spectrum resources that are not used by the primary user (PU) and allowed them to opportunistically be accessed without causing harmful interference to the PU [3], which is known as cognitive wireless sensor networks (CWSNs) [4].

In a CWSN, cooperative spectrum sensing (CSS) is the most critical technology, which refers to multiple sensors using local sensing technology to detect the signal of the PU, and then submitting the sensing information to the fusion center (FC). The FC makes the final decision on the status of the PU through a specific rule. Therefore, those sensors are allowed to opportunistically access the channel being underutilized by the PU according to the final decision regarding the presence of the phenomenon. However, the cooperative process also provides malicious sensor nodes an opportunity to falsify sense information after completing local sensing and submit it to the FC [5], misleading the FC to make an incorrect global decision and selfishly occupying spectrum resources or causing excessive interference to the PU [6]. This is called a Byzantine attack [7], and there is already a lot of research work in this field.

1.2. Related Works and Motivation

A Byzantine attack in CSS, a.k.a. spectrum sensing data falsification (SSDF) attack, is one of the key adversaries to the success of CR networks. In recent years, research on Byzantine attacks and defense strategies has attracted attention and become a hot topic. In [8], L. Zhang et al. conducted extensive research and in-depth analysis of Byzantine attack and defense for CSS in CR networks. In [9], J. Wu et al. analyzed strategies between Byzantine attack and the FC in CSS for CR networks and further evaluated the safety factor from the Byzantine attacker’s perspective using a trust-value algorithm [10]. In [11], K. Zeng et al. presented a secure CSS scheme based on a reputation accumulation mechanism to mitigate the adverse impact of misbehaved CRs. A secure CSS strategy based on a reputation mechanism for CWSNs is proposed by X. Luo to defend against Byzantine attacks [12]. In [13], R. Chen et al. proposed a weighted sequential probability ratio test (WSPRT) against Byzantine attack to improve the robustness of data fusion, which integrates reputation accumulation of [11] into a sequential probability ratio test (SPRT) and reduces the data collection communication overhead at the FC. However, these studies did not consider the situation of many Byzantine attackers, which would completely make the FC blind, resulting in the failure of these reputation methods based on global decisions. In the presence of massive Byzantine attackers, Z. Sun et al. proposed a scheme in which the reliability value is used for dynamic selection of the sensing strategy between CSS and independent spectrum sensing [14,15]. However, the authors did not mitigate the negative impact of massive Byzantine attackers but instead replaced CSS with independent spectrum sensing. R. Lin et al. made use of blockchain to prevent independent and cooperative Byzantine attacks from malicious vehicle users [16]. It is obvious that this decentralized method to resist Byzantine attacks is inefficient. In [17], Y. Fu et al. proposed a scheme to identify probabilistic Byzantine attackers using consistent properties of each sensing user’s historically reported data. This consistency property method has certain requirements for the scenario of the primary network.

Further, Z. Li et al. studied the sequential binary hypothesis test problem with Byzantine agents in both the FC and fully distributed formulations [18]. In [19], an estimation diffusion least-mean-square-H (H represents the average node degree of all nodes) algorithm was proposed by F. Wan et al. to achieve better sensing performance against Byzantine and manipulation attacks. In [20], C. Quan et al. investigated the negative effect of Byzantine attacks on CSS performance and used an ordered transmission scheme to solve the binary hypothesis test problem. Furthermore, in [21,22], the authors proposed reputation and audit-based clustering with/without auxiliary anchor node algorithms against Byzantine attacks in wireless sensor networks. These Byzantine defense algorithms and strategies exhibit high computational complexity and necessitate the availability of specific prior information (strategy analysis of both the FC and Byzantine agents) or ideal assumptions (auxiliary anchor node); however, their applicability is limited to a certain extent.

In [23], a Dempster–Shafer evidence theory-based CSS for CWSNs was introduced by D. Yao et al. to deal with Byzantine attacks. M. Ridouani et al. proposed an authentication process and shifted spectrum sensing process to enhance the detection performance [24]. Considering an erroneous feedback channel, a single decision reporting algorithm was developed by A. Chouhan et al. to mitigate the effect of a Byzantine attack [25]. Furthermore, the authors proposed a machine learning technique [26] to identify malicious users in a CR network using the principal component analysis algorithm. To resist the dominated cooperative probabilistic Byzantine attack, L. Chen et al. proposed a JS-divergence-based reputation algorithm to identify Byzantine attackers [27]. In [28], A. Parmar et al. used a Gaussian mixture model to formulate an anomaly detection algorithm for detecting Byzantine attacks. Although the above defense algorithms can identify and suppress Byzantine attacks and secure CSS to some extent, they still cannot effectively defend against large-scale attacks (with a high attack ratio) and low-intensity attacks (with a low attack probability) and ensure the performance and efficiency of CSS in CWSNs.

Additionally, machine learning has gradually become one of the most effective methods for solving network security problems in recent years. In [29], one class supporting vector machine (SVM) to obtain a modified SPRT was used by J. Parras and S. Zazo to not only detect a Byzantine attack but potentially any other attack mechanism that does not have a similar spectrum to the expected signal from normal sensors. N. Marchang et al. investigated a series of machine learning techniques for Byzantine identification in [30,31]. Z. Luo et al. proposed an influence-limiting defense against learning evaluation-beating attacks or other similar attacks [32]. Moreover, by providing a maximum margin hyperplane [33], Z. Zhang et al. used SVM to identify Byzantine attackers, where the generated spectrum sensing data features benefit from the PU status during the training process. Though these machine learning algorithms have certain advantages in prior information, performance, and application scenarios, achieving these goals requires massive training sequences and computational costs, which are not suitable for low-energy sensors. In large-scale CWSNs, in particular, lightweight computing and efficiency for CSS are indispensable.

1.3. Our Contributions

In order to guarantee the premise of CWSNs, the CSS process should be carefully taken into consideration in the context of Byzantine attacks. Therefore, a widely applicable Byzantine attack model should be developed first, and considering the robustness of subsequent Byzantine identification and suppression algorithms, the attack probability and attack ratio should not be limited. Also, we considered the sequential binary hypothesis test to improve CSS efficiency and beta distribution function (BDF) to guarantee CSS performance. The contributions of our work can be summarized as:

To characterize Byzantine behavior from malicious sensor nodes in the CSS process, we design a pair of flexible attack parameters, i.e., attack probability and attack ratio, to carry out various attack strategies (i.e., always attack, probabilistic attack) and attack scales (i.e., small/large-scale attack).

Further, we convert the binary hypothesis test problem on the phenomenon of the presence of the PU into the sequential process to realize CSS among multiple sensors. To accurately evaluate the reputation value of each sensor, we formulate a beta reputation model to identify malicious sensor nodes.

Considering the storage capacity of the FC, the positive and negative total evaluation within a window period is designed to quantify reliable and unreliable sensor nodes. Moreover, we exploit tuning parameters that reflect linear and multiplier growth to accurately determine the weight of the likelihood ratio in the sequential process.

1.4. Organization

The rest of this paper is organized as follows. In Section 2, a CWSN and Byzantine attack model are presented. Section 3 formulates BDF for CSS to defend against Byzantine attacks from malicious sensor nodes. The correctness and effectiveness of BDF are verified in Section 4 in terms of the error probability and the average number of samples. Finally, Section 5 concludes this paper.

2. System Model

In this section, we propose a centralized CWSN model in the presence of malicious sensor nodes. On this basis, we further model a CSS paradigm among multiple sensor nodes in which malicious sensor nodes launch Byzantine attacks, including various attack strategies and attack scales.

2.1. CWSN Model

We consider a centralized CWSN consisting of a PU, an FC, and $N$ collaborative sensors, where the proportion of malicious sensor nodes is $\rho$, as shown in Figure 1. Since our research relies on the FC, it is only applicable to a centralized network and not to distributed networks that do not rely on the FC. In a spectrum sensing frame structure [34], a frame duration includes a sensing slot, a reporting slot, and a data transmission slot, as shown in Figure 2. To opportunistically access the available channel and avoid harmful interference to the normal communication of the PU, all sensors detect the PU’s signal with the help of the local sensing technology at a sensing slot. Then, at the reporting slot, each sensor individually submits its own sensing result to the FC via the error-free common control channel. After receiving the sensing results, the FC needs to make a global decision about the PU’s status using a specific rule. Finally, based on the global decision, the FC broadcasts a message to the sensors, which determines whether sensor nodes are allowed to access the channel. In detail, if the global decision is 1, sensors will be allowed to transmit data at the data transmission slot, otherwise, they will be forbidden from accessing the channel and continue spectrum sensing at the next sensing frame.

There may be various types of PUs and low computational complexities in CWSNs, and energy detection (where the energy value is the mean square of the received signal within a certain sampling time) is often used as a local spectrum sensing technique to detect the PU signal. Then, by comparing the energy value with a pre-set decision threshold, a local binary decision about the presence or absence of the PU can be made. This is the process of local spectrum sensing. A pair of local spectrum sensing performances, including the local false alarm and miss-detection probabilities, are denoted by $P_f$ and $P_m$, respectively.

2.2. Byzantine Attack Model

On the basis of the CSS paradigm and its open nature, some malicious nodes may take advantage of this opportunity to launch Byzantine attacks. Specifically, after completing the local sensing, the malicious sensor nodes will falsify their own sensing result and submit it to the FC, aiming to mislead the FC into making an incorrect decision about the PU’s status. To further analyze Byzantine behaviors, the local false alarm and miss-detection probabilities are assumed to be the same for each sensor node. Then, at the $k$-th sensing, the local sensing result, and the received sensing result are denoted by $L_i\left(k\right)$ and $R_i\left(k\right)$, respectively. Byzantine behaviors from the malicious sensor node can be described by an attack probability as

$$\alpha =\left\{\begin{array}{c}P\left(R_i\left(k\right)=1|L_i\left(k\right)=0\right)\\ P\left(R_i\left(k\right)=0|L_i\left(k\right)=1\right)\end{array} \right.$$

(1)

where $\alpha$ varies from 0 to 1. Such an attack strategy represents the attack probability that the malicious sensor nodes flips the sensing result from 1 to 0 or from 0 to 1. Therefore, the false alarm and miss-detection probabilities at the malicious sensing nodes can be given by

$$P_{fa}=\left(1-P_f\right)\alpha +P_f\left(1-\alpha \right)=\alpha +P_f-2\alpha P_f$$

(2)

$$P_{ma}=\left(1-P_m\right)\alpha +P_m\left(1-\alpha \right)=\alpha +P_m-2\alpha P_m$$

(3)

Specifically, if the attack probability $\mathsf{\alpha }$ is set to 1, the malicious sensor node always carries out the attack strategy. In addition to the attack strategy, the proportion of malicious sensor nodes should also be taken into consideration. Many current research works only consider the situation where there are relatively few malicious nodes, because the global decision at the FC cannot be distorted by a small number of malicious sensor nodes, so malicious sensor nodes can be easily identified using global decisions. Once there are a large number of malicious sensor nodes in CWSNs, i.e., $\rho >50\%$, traditional algorithms will no longer be effective. Therefore, in order to achieve a more secure CSS algorithm, defense against large-scale attacks should also be indispensable.

3. Beta Distribution Function

In this section, encouraged by SPRT, we adopt SPRT as the fundamental CSS framework to reduce the number of samples required by the FC. On this basis, a beta reputation model is formulated to distinguish malicious sensor nodes and integrated into the sequential CSS process.

3.1. Sequential Process

Based on the received sensing results, the FC adopts a specific fusion rule to mitigate the negative impact of Byzantine attacks on CSS. In addition, considering the number of sensors, the cooperative efficiency should be also taken into consideration to decrease the samples required by the FC, therefore reducing the communication overhead from the sensor node to the FC. To this end, we propose BDF for CSS against Byzantine attacks.

In the beginning, we adopted SPRT as a CSS framework to improve the cooperative efficiency when the FC collects the sensing information from sensor nodes. In detail, SPRT requires the knowledge of a priori probabilities of $L_i\left(k\right)$ when the hypotheses ${\mathcal{H}}_0$ or ${\mathcal{H}}_1$ of the PU is absent or present, i.e., $P_r\left(L_i\left(k\right)|{\mathcal{H}}_0\right)$ and $P_r\left(L_i\left(k\right)|{\mathcal{H}}_1\right)$. Relying on these assumptions, the FC sequentially calculates the likelihood ratio and makes a global decision according to the following decision variable:

$$S_n={\displaystyle \prod }_{i=1}^n{\displaystyle \frac{P_r\left(L_i\left(k\right)|{\mathcal{H}}_1\right)}{P_r\left(L_i\left(k\right)|{\mathcal{H}}_0\right)}}$$

(4)

where $n$ represents the number of sensing results/samples required by the FC at a sensing frame and varies from 1 to $N$. Further, the global decision will be made according to the following criterion

$$\left\{\begin{array}{c}{S}_n\ge {\xi }_u, the FC accepts {\mathcal{H}}_1\\ S_n\le {\xi }_l, the FC accepts {\mathcal{H}}_0\\ {\xi }_l<S_n<{\xi }_u, the FC takes next observation\hfill \end{array} \right.$$

(5)

where ${\xi }_u$ and ${\xi }_l$ are defined as

$${\xi }_u={\displaystyle \frac{1-{\overline{P}}_f}{{\overline{P}}_m}},$$

(6)

and

$${\xi }_l={\displaystyle \frac{{\overline{P}}_f}{1-{\overline{P}}_m}},$$

(7)

${\overline{P}}_f$ and ${\overline{P}}_m$ are tolerated false alarm and miss-detection probabilities, respectively. It should be noted that even though it has been shown that the SPRT terminates with probability one, the number of samples $n$ required for terminating the sequential process is determined by $P_r\left(L_i\left(k\right)|{\mathcal{H}}_0\right)$, $P_r\left(L_i\left(k\right)|{\mathcal{H}}_1\right)$, ${\xi }_l$ and ${\xi }_u$, therefore it is random [35].

Compared to Bayesian detection and Neyman–Pearson, SPRT to a certain extent saves the number of samples required for the FC to efficiently make a global decision without any performance loss. However, similar to Bayesian detection and Neyman–Pearson, it also cannot defend against Byzantine attacks. Therefore, it is necessary to take security and performance into consideration. Motivated by this disadvantage of SPRT, a reputation mechanism is usually integrated into the weighted sequential process, as given in

$$\begin{gathered} S_n={\displaystyle \prod }_{i=1}^n{\left[{\displaystyle \frac{P_r\left(L_i\left(k\right)|{\mathcal{H}}_1\right)}{P_r\left(L_i\left(k\right)|{\mathcal{H}}_0\right)}}\right]}^{w_i\left(k\right)} \\ ={\displaystyle \prod }_{i=1}^n{\left\{{\left[{\displaystyle \frac{P_r\left(L_i\left(k\right)=1|{\mathcal{H}}_1\right)}{P_r\left(L_i\left(k\right)=1|{\mathcal{H}}_0\right)}}\right]}^{d_i\left(k\right)}\ast {\left[{\displaystyle \frac{P_r\left(L_i\left(k\right)=0|{\mathcal{H}}_1\right)}{P_r\left(L_i\left(k\right)=0|{\mathcal{H}}_0\right)}}\right]}^{1-d_i\left(k\right)}\right\}}^{w_i\left(k\right)} \\ ={\displaystyle \prod }_{i=1}^n{\left\{{\left[{\displaystyle \frac{1-P_m}{P_f}}\right]}^{d_i\left(k\right)}\ast {\left[{\displaystyle \frac{P_m}{1-P_f}}\right]}^{1-d_i\left(k\right)}\right\}}^{w_i\left(k\right)} \end{gathered}$$

(8)

To be specific, the reputation of the $i$-th sensor nodes after the $k$-th sensing is obtained by following reputation mechanism [6]

$$T_i\left(k\right)=T_i\left(k-1\right)+{\left(-1\right)}^{R_i\left(k\right)+g\left(k\right)}$$

(9)

Thus, the likelihood ratio weight of the $i$-th sensor node is designed as [8]

$$w_i\left(k+1\right)=\left\{\begin{array}{c}0, T_i\left(k\right)\le -g\\ {\displaystyle \frac{T_i\left(k\right)+g}{\max \left(T_i\left(k\right)\right)+g}}, T_i\left(k\right)>-g\end{array} \right.$$

(10)

However, when the FC makes a global decision, WSPRT is easily affected by large-scale attacks and cannot accurately output reputation values, leading to the failure of weight. Hence, a reputation mechanism remains a major challenge. To this end, we integrate a beta reputation model into the sequential process in the following subsection.

3.2. Beta Reputation Model

Following the above purpose, we consider that during the local sensing, the sensing results can be categorized into two possibilities, following a binomial distribution. The beta distribution is suitable for describing the probability distribution characteristics of binomial events. Hence, we can establish a reputation model to allocate member reputation values. This approach helps mitigate reputation fluctuations caused by factors such as noise uncertainty, multipath fading, or shadowing, thereby safeguarding the normal senor nodes from excessive defensive actions by the FC.

Let $x$ represent the actual local sensing result transmitted by sensors and $y$ represent the local sensing result after being falsified by malicious sensors. Use $r$ to denote the number of occurrences of event $x$ and $s$ to represent the number of occurrences of event $y$. By setting ${\beta }_0=r+1$ and ${\beta }_1=s+1$, the probability density function can be derived by [7]

$$f\left(p|{\beta }_0,{\beta }_1\right)={\displaystyle \frac{\mathit{\Gamma }\left({\beta }_0+{\beta }_1\right)}{\mathit{\Gamma }\left({\beta }_0\right)\mathit{\Gamma }\left({\beta }_1\right)}}{p}^{{\beta }_0-1}{\left(1-p\right)}^{{\beta }_1-1}$$

(11)

where $\mathit{\Gamma }\left(·\right)$ is a Gamma function, $p$ represents the probability of sensing behaviors and $0\le p\le 1$. ${\beta }_0>0$, ${\beta }_1>0$. The expectation of event $x$ is

$$E\left[Beta\left({\beta }_0,{\beta }_1\right)\right]={\displaystyle \frac{{\beta }_0}{{\beta }_0+{\beta }_1}}$$

(12)

Then, the reputation value of the $i$-th sensor node is evaluated using the beta function, i.e.,

$$T_i=Beta\left(r_i+1,s_i+1\right)={\displaystyle \frac{r_i+1}{r_i+s_i+2}}$$

(13)

3.3. Weight Evaluation

To effectively combat Byzantine attacks, the FC employs a beta reputation system to allocate reputation values to sensor nodes based on historical sensing behavior. This approach establishes a dynamic reputation evaluation for CSS by means of a data delivery mechanism. It is known that a fixed frame duration consists of a sensing slot, a reporting slot, and a data transmission slot in a periodic spectrum sensing frame. After the reporting slot, the FC makes a global decision about the PU status according to (2), such as, when the global decision is 1, all sensors are forbidden from accessing the channel and while the global decision is 0, all sensors are allowed to access the channel. Nevertheless, due to the negative impact of a Byzantine attack, the global decision may be unreliable. Hence, we have the following considerations: (a) if global decision 1 is incorrect, then the original channel must be idle, and malicious sensor nodes selfishly occupy the channel by tampering with the sensing results, (b) if global decision 0 is incorrect, then the original channel must be busy, and malicious sensor nodes must have interfered with the PU by tampering with the sensing results.

This data delivery mechanism can be used by the FC to determine the reliability of the global decision in the current frame and to measure the local sensing results without being affected by the Byzantine attack. We label it as $G\left(k\right)$ and use it as a standard to measure the reliability of local sensing results. By comparing the local sensing results with $G\left(k\right)$, the result deviation at the $k$-th sensing can be given by

$$D_i\left(k\right)=\sqrt{{\displaystyle \frac{{\left(L_i\left(k\right)-G\left(k\right)\right)}^2}{N}}}$$

(14)

Then, the average value of the sensing result deviation is

$$\overline{D_i}\left(k\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum }_{i=1}^N{D}_i\left(k\right)$$

(15)

Further, let $e_i\left(k\right)$ represent the positive and negative evaluations of the sensing results obtained by the $i$-th sensor. $\overline{D_i}\left(k\right)\ge D_i\left(k\right)$ indicates that the local sensing results of the $i$-th sensor at the $k$-th sensing are reliable, and a positive evaluation $e_i\left(k\right)=1$ is assigned to the $i$-th sensor. Conversely, if the local sensing result is deemed unreliable, it is assigned a negative evaluation $e_i\left(k\right)=0$.

Considering the storage capacity of the FC, we assume that the historical evaluation values of the sensor is $L$, which means the FC retains the evaluation values derived from $L$ instances of sensing results. Thus, the positive total evaluation $p_i\left(k\right)$ and negative total evaluation $n_i\left(k\right)$ of the $i$-th sensor at the $k$-th sensing are computed as

$$p_i\left(k\right)=\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^L}{e}_i\left(k\right){\left(w_i\left(k\right)\right)}^{L-n}, if k<L\\ {\displaystyle \sum _{j=k-L+1}^L}{e}_i\left(k\right){\left(w_i\left(k\right)\right)}^{L-n}, otherwise\end{array}\right.$$

(16)

and

$$n_i\left(k\right)=\left\{\begin{array}{c}{\displaystyle \sum _{j=1}^L}(1-e_i\left(k\right)){\left(w_i\left(k\right)\right)}^{L-n}, if k<L\\ {\displaystyle \sum _{j=k-L+1}^L}(1-e_i\left(k\right)){\left(w_i\left(k\right)\right)}^{L-n}, otherwise\end{array} \right.$$

(17)

respectively.

The total evaluation is then incorporated into the beta reputation system. Specifically, for the $i$-th sensor during the $k$-th sensing, the accumulated evaluation values are considered. The reputation parameters $r_i\left(k\right)$ and $s_i\left(k\right)$ are expressed as

$$r_i\left(k\right)=p_i\left(k\right)+{\varphi }_r$$

(18)

$$s_i\left(k\right)=n_i\left(k\right)\ast {\varphi }_s$$

(19)

where ${\varphi }_r$ and ${\varphi }_s$ represent tuning parameters that reflect linear and multiplier growth of different types of sensing results, respectively.

Finally, the reputation value/weight of the $i$-th sensor after the $k$-th sensing can be expressed as

$$T_i\left(k+1\right)=w_i\left(k+1\right)={\displaystyle \frac{p_i\left(k\right)+{\varphi }_r+1}{p_i\left(k\right)+n_i\left(k\right)\ast {\varphi }_s+{\varphi }_r+2}}$$

(20)

In addition, the higher the reputation value of the sensor, the higher the sensing result will be prioritized in the likelihood ratio at the next sensing, thereby further improving the efficiency of collaboration. The whole process of BDF is illustrated in Figure 3.

4. Simulation Results

In this section, simulation results are presented to demonstrate the effectiveness of the proposed BDF by comparing it with SPRT and WSPRT in terms of the error probability $Q_e$ and the average number of samples $N_{ave}$. In a sensing observation period (2000 sensing frames), the attack probability $\alpha$ varies from 0 to 1 at an interval of 0.02, and the attack probability $\rho =\left[0.1, 0.2, 0,3\right]$ and $\rho =\left[0.7, 0.8, 0,9\right]$ are presented to characterize the small/large-scale attack, respectively. The values for other simulation parameters are set as follows: The number of sensors $N$ participating in CSS is 100. Regardless of whether the sensor nodes are malicious or normal, their local detection probability $P_d$ and local false alarm probability $P_f$ are both 0.7 and 0.3, respectively, assuming that the probability $P\left({\mathcal{H}}_0\right)$ of the PU’s status ${\mathcal{H}}_0$ is 0.1 and the probability $P\left({\mathcal{H}}_1\right)$ of the PU’s status ${\mathcal{H}}_1$ is 0.9. In addition, in the sequential process, the tolerated miss-detection probability ${\overline{P}}_m$ and the tolerated false alarm probability ${\overline{P}}_f$ are set to 10⁻³ and 10⁻⁴, respectively. The reputation threshold $g$ of WSPRT is 5.

4.1. Always Attack

The always attack is a common attack strategy that is often considered in many Byzantine identification and suppression algorithms. Therefore, we consider the error probability and the average number of samples in the context of this attack strategy. The attack ratio $\rho$ varies from 0 to 0.8 at an interval of 0.02.

As illustrated in Figure 4, when those malicious sensor nodes launch always attack, the error probability of SPRT starts to shake when the attack probability is 0.3 and increases to 1 when the attack probability exceeds 0.5. At the same time, the error probability of WSPRT performs very well before the attack ratio reaches 0.5, but once $\rho$ exceeds 0.5, it also shakes up to 1 and remains stable. In contrast, the proposed BDF consistently maintained very accurate PU detection. The above results strongly confirm that both SPRT and WSPRT are negatively affected by the blind problem to varying degrees.

Following the error probability in the context of always attack, the average number of samples is presented in Figure 5. It can be seen from Figure 5 that when the attack probability is low, because of the sequential idea, all three fusion rules only require a small number of samples to accurately detect the PU. However, as the attack probability increases, the FC of SPRT and WSPRT begins to require more samples to make a global decision (although the global decision is not accurate at this time), and the required number of samples increases first and then decreases (constantly shaking during the process) because of the randomness of the sequential process (samples may come from malicious or normal sensor nodes) and the unreliability of global decision. Finally, since the FC is distorted, the performance is significantly degraded and the number of samples does not need to be greater. In contrast, the BDF sample size remains unchanged at around 5.

Although always attack is a common attack strategy and many algorithms have paid attention to it, they only consider situations with lower attack ratios, so always attack is also easy to identify and suppress. Here, we not only consider always attack but also various attack ratios. However, from the above simulation results, our proposed BDF exhibits excellent performance in both the error probability and sample size during the sequential process due to considering the data delivery mechanism and adapting the corresponding weight evaluation.

4.2. The Error Probability

In order to get an insight into a small/large attack’s influence on the error probability, we compare the error probabilities of SPRT, WSPRT, and BDF under various attack probabilities in the context of a small/large attack.

4.2.1. Small-Scale Attack

As shown in Figure 6, regardless of the attack probability adopted by malicious sensor nodes, SPRT, WSPRT, and BDF in the context of small-scale attacks can basically achieve 100% detection accuracy for the PU signal because even SPRT has a certain level of Byzantine fault tolerance. From this, it can be seen that under a small-scale attack, even always attack cannot make the FC blind (the Byzantine attack makes the global decision of the FC not more accurate than random guesses). It should be noted that since SPRT does not have any defense capabilities to resist Byzantine attacks, once the attack ratio increases, i.e., when $\rho =\left[0.2, 0,3\right]$, the error probability also increases, to some extent.

4.2.2. Large-Scale Attack

Different from small-scale attacks, the negative impact of large-scale attacks on the error probabilities of SPRT, WSPRT, and BDF is more significant. As illustrated in Figure 7, the error probabilities of SPRT, WSPRT, and BDF remain zero when the attack probability is less than 0.2. However, as the attack probability increases, the three methods exhibit different variations of the error probability. In detail, the error probability of SPRT fluctuates upwards; the larger the attack ratio, the higher the error probability, until it stabilizes. WSPRT suppresses the Byzantine attack to a certain extent through the weight of likelihood ratio based on SPRT, presenting the following differences: (1) the performance of WSPRT deteriorates slower than that of SPRT under the same attack strategy before the FC is blind; (2) once the FC is blind, the error probability jitter of WSPRT is more severe than that of SPRT; (3) the attack probability further increases and the error probability of WSPRT is also higher than that of SPRT. This is because the ratio and probability of attack gradually increase, making the global decision of the FC unreliable. WSPRT relies on the global decision to measure the reliability of the local sensing result and design weight, resulting in unstable performance once the global decision is incorrect, and, coupled with the random calculation of likelihood ratio, ultimately worse performance (normal sensor nodes are considered malicious, and vice versa).

In contrast to SPRT and WSPRT, BDF shows significant performance at $\rho =\left[0.7, 0,8\right]$, indicating that the beta reputation model can accurately identify malicious sensor nodes; that is to say, the reputation value is accurately evaluated and not affected by the accuracy of the global decision. Based on this, the positive and negative total weight evaluation accurately amplifies and reduces the positive and negative impacts on the global decision-making of the FC, respectively. Further, when $\rho =0.9$, the error probability of BDF increases (first increases and then decreases), which also indicates that always attack is relatively easy to overcome; however, overall, its error probability is still not higher than 0.12.

4.3. The Average Number of Samples

The cooperative performance of SPRT, WSPRT, and BDF is presented by means of the error probability in the context of small/large-scale attacks. The next thing that should be considered is to evaluate the cooperative efficiency using the average number of samples.

4.3.1. Small-Scale Attack

The average numbers of samples for SPRT, WSPRT, and BDF in the context of small-scale attacks are simulated in Figure 8. In the case of $\rho =\left[0.1, 0.2, 0,3\right]$, no matter what attack probability the malicious sensor node adopts, it cannot make the FC blind. However, due to the malicious sensor node tampering with the sensing result, the sequential SPRT and WSPRT processes randomly calculate the likelihood ratio, so the required number of samples is also random and less likely to make the decision variable ${\Lambda }_n$ satisfy the upper and lower threshold conditions. Moreover, as the attack probability increases, the fluctuation of the average sample size also increases.

Unlike SPRT and WSPRT, the number of samples required for BDF never exceeds 5.5. On the one hand, BDF accurately identifies malicious sensor nodes through a data delivery mechanism, and on the other hand, the beta reputation model enables the FC to rely on normal sensor nodes to the greatest extent possible while suppressing malicious sensor nodes. In addition, BDF can prioritize the sensing results of sensors with high credibility, enabling the FC to make global decisions quickly and accurately.

4.3.2. Large-Scale Attack

The sample size under a large-scale attack appears to be more regular, as at a certain attack probability, the Byzantine attack may make the FC blind, as shown in Figure 9. Firstly, the sample sizes of the three methods increase and then decrease with the increase in attack probability, except WSPRT has the largest amplitude. Undoubtedly, the increase in attack probability may make the global decision less reliable, but the weight of WSPRT makes the global decision more unreliable and random. In order to ensure that the decision variable meets the upper and lower thresholds in the sequential process, the FC has to require more samples. Once the FC is blind, it indicates that the global decision is completely unreliable, and the FC only needs a small number of samples to make an incorrect global decision.

The above analysis and discussion of BDF are conducted in a serial case. If there are fewer sensors in parallel, the difference between serial and parallel efficiencies is not significant. However, when there are more sensors, serial efficiency is better, reducing the computational load of the FC as well as unnecessary overhead communication between sensors and the FC. In addition, if considered properly, there is still room to improve the efficiency of serial BDF. For example, the PU’s status can be characterized based on the local decision changes in the front and rear sensors rather than the local decision itself; that is, the front and rear differences can be used to further improve efficiency.

5. Conclusions and Future Work

In this paper, we investigated Byzantine attacks on CSS in CWSNs. To mitigate the negative impact of Byzantine attacks on CSS, we propose BDF, which includes a sequential process, beta reputation model, and weight evaluation. In BDF, we are motivated by SPRT to integrate the beta reputation function into a weight-for-weight sequential process and sequentially calculate the likelihood ratio in a descending reputation order. Finally, in contrast to SPRT and WSPRT, a series of numerical simulation results demonstrate the correctness and effectiveness of the proposed BDF regarding the error probability and the average number of samples under small/large-scale attacks and attack probabilities.

There are many interesting questions that remain to be explored in the future, such as a Byzantine attack strategy based on soft combining. Additionally, the sequential idea may be further extended to machine learning methods to ensure the accuracy, security, and efficiency of CSS.