## 2 System Model  
We consider the operations of a CR network where \(N\) SUs try to transmit data to a data center (DC) through \(K\) unlicensed channels. Let \(\mathcal{N}\) and \(\mathcal{K}\) denote the sets of SUs and channels, respectively. Primary users (PUs) that are licensed to use channels are assumed to operate in a time-slotted model. In this paper, we assume that the operation of a PU in channel \(k\) follows the Markov chain model. The operation can be expressed by the state transition probability between two states of the PU as \(P_{PA}^{k}=\mathbb{P}(s_{(l+1)}^{k}=P|s_{(t)}^{k}=A)\) and \(P_{AA}^{k}=\mathbb{P}(s_{(t+1)}^{k}=A|s_{(t)}^{k}=A)\) where the symbols (\(P\)) and (\(A\)) represent the presence and absence of the PU, as shown in Figure 1. The transition probabilities of the PU from state \(P\) to state \(A\) and from state \(A\) to itself are defined as \(P_{PA}^{k}=\mathbb{P}(s_{(l+1)}^{k}=P|s_{(t)}^{k}=A)\) and \(P_{AA}^{k}=\mathbb{P}(s_{(t+1)}^{k}=A|s_{(t)}^{k}=A)\), respectively, where \(s\) is the state of the PU on channel \(k\), and \(t\) is the index of the time slot. In the network, \(E\) eavesdroppers will try to overhear data from the SUs as shown in Figure 2. Let \(\mathcal{V}\) denote the set of eavesdroppers in the network.  
At the beginning of each time slot, the SU selects a channel for sensing where energy detection is used, and then, if the channel is free, the SU will transmit data over the channel, which is assumed as an additive white Gaussian noise (AWGN) channel; otherwise, the SU is not allowed to access the channel, then it will wait for the next time slot and repeat the process.  
Figure 1: Markov chain states of the PU.  
Figure 2: The system model.  
When SU \(i\) transmits data over available channel \(k\), the received signal-to-noise ratio (SNR) can be given as:  
\[\rho_{i,k}=\frac{h_{i,k}P_{i,k}}{\sigma^{2}+\sum\limits_{j\in N_{k}\backslash\{i\} }h_{j,k}P_{j,k}} \tag{1}\]  
where \(\sigma^{2}\) is the noise variance, \(h_{i,k}\) is the channel gain of SU \(i\) on channel \(k\), \(P_{i,k}\) is the transmission power of SU \(i\) over channel \(k\), and \(N_{k}\) is the set of SUs that transmit data over channel \(k\).  
Accordingly, let us define the achievement of SU \(i\) over channel \(k\) as follows:  
\[R_{i,k}=\log(1+\rho_{i,k}). \tag{2}\]  
While SU \(i\) transmits data over channel \(k\), an eavesdropper may try to listen to the data over the channel. Then, the received SNR at the EV can be given as follows:  
\[\rho_{i,k}^{e}=\frac{g_{i,e,k}P_{i,k}}{\sigma^{2}+\sum\limits_{j\in N_{k} \backslash\{i\}}g_{j,e,k}P_{j,k}} \tag{3}\]  
where \(g_{i,e,k}\) is the channel gain at eavesdropper \(e\) when it bears the data of SU \(i\) over channel \(k\).  
The achievement of eavesdropper \(e\) can also be given as follows:  
\[R_{i,k}^{e}=\log(1+\rho_{i,k}^{e}). \tag{4}\]  
The goal of the SU is to maximize its own achievement and minimize the achievement of eavesdroppers. To measure the performance of the proposed scheme, in the paper, we utilize the secrecy rate of the SU, defined as the difference between the expected reward (i.e., throughput) of both the SU and that of EVs, since the secrecy rate is often used as the main performance metric for anti-eavesdropping schemes [25; 26]. The secrecy rate of SU \(i\) can be given as follows:  
\[S_{i,k}=\left(\frac{R_{i,k}-\max\limits_{e}R_{i,k}^{e}}{R_{i,k}}\right)^{+} \tag{5}\]  
where the function \((a)^{+}\) is given as \(\left(a\right)^{+}=\max\left(a,0\right)\).