## 3 The Proposed Key Generation Protocol
### Channel Probing  
Channel probing is a critical process that enables both communicating parties to derive the random physical source of the channel. Ideally, both parties should transmit probing signals simultaneously through full-duplex probing (as illustrated in Figure 3). In this case, the channel state is extracted by both parties at the same time \(t_{0}\), ensuring the highest correlation in their probing results. However, in a half-duplex scenario, [PERSON] and [PERSON] probe the channel state at different times, \(t_{1}\) and \(t_{2}\), respectively. During an interval \(|t_{2}-t_{1}|\), factors such as node movement or environmental changes may cause slight variations in the channel, thereby reducing the correlation of the probing results. In the following, we discuss the challenges and solutions associated with channel probing in the AIS. The MAC protocol used by Class A AIS equipment is Self-Organized Time Division Multiple Access (SOTDMA). In the subsequent discussion on key generation, we base our analysis on this protocol. Since nodes in a TDMA system take turns occupying channels in different time slots, simultaneous data transmission by both communicating parties in the same time slot results in signal conflicts. Therefore, full-duplex probing is not practical in AIS communication. Consequently, [PERSON] and [PERSON] can only send probing signals in different time slots. The gain of a wireless communication channel varies due to changes in time or space. The correlation time of a channel is defined as the time interval during which the channel's characteristics remain relatively constant (i.e., the channel gain does not change significantly). This time interval is influenced by the Doppler frequency shift, which arises from the relative motion between communication nodes. The correlation time for a Rayleigh fading channel can be computed using the autocorrelation function of the channel [29]. The autocorrelation function \(R(\tau)\) of the channel gain is given by  
\[R(\tau)=J_{0}(2\pi f_{d}\tau) \tag{6}\]  
This formula represents the correlation of the channel gain as a function of the time delay \(\tau\). The Doppler frequency shift \(f_{d}\) can be calculated from the relative velocity \(\Delta v\) between the communication nodes:  
\[f_{d}=\frac{\Delta v}{c}f_{VHF}\cos\theta \tag{7}\]  
where \(J_{0}\) is the 0-th order Bessel function, \(c\) is the speed of light, \(\theta\) is the angle between the velocities of the two ships, and \(f_{VHF}\) is the VHF frequency. Since AIS 1 is the preferred  
Figure 3: (a) Full-duplex probing and (b) half-duplex probing. channel, we use \(f_{VHF}=161.975\) MHz in the subsequent calculations. The channel correlation time \(t_{c}\) is defined as the time delay \(\tau\) at which \(R(\tau)\) decays to a specific threshold, typically 0.5. When the function value reaches 0.5,  
\[t_{c}\approx\frac{0.423}{f_{d}} \tag{8}\]  
In this case, [PERSON] and [PERSON] must complete their channel probing within the correlation time \(t_{c}\). Otherwise, the significant disparity in their probing values greatly increases the cost of subsequent information coordination. To ensure that the protocol is applicable under various conditions and to facilitate discussion, we set \(\cos\theta=1\) when solving for \(t_{c}\). This represents a more stringent scenario, where the channel undergoes the fastest changes. Under these conditions, the relationship between the correlation time \(t_{c}\) and the relative speed is as illustrated in Figure 4.  
We note that the AIS frame length is 60 s, divided into 2250 time slots, each lasting 26.67 ms--just enough to transmit a 256-bit standard AIS message. Therefore, the minimum duration of a round of probing using an AIS message is two time slots. However, when the relative speed \(\Delta v>28\), knots, the relative time interval \(t_{c}\) between the two parties becomes shorter than two time slots (see Figure 4). This results in the probing results of the two parties in a single round of probing being uncorrelated.