## 3 Development of an Imitation-Resistant Zero-Knowledge Authentication Protocol Based on an MPPC
The analysis of Figure 4 shows that the use of an error-correction MPPC can improve the noise resistance of the SRSU authentication system. So, at the signal-to-noise ratio equal to 8 dB, the authentication system without the use of an error-correction code provides a bit error rate (BER) equal to \(BER_{1}=2\times 10^{-4}\). The use of the BCH code provides a BER equal to \(BER_{2}=2\times 10^{-5}\). The application of the developed error correction algorithm for the MPPC allows for reducing the BER down to \(BER_{3}=2\times 10^{-6}\). In order to provide \(BER=10^{-5}\), it is necessary that the signal-to-noise ratio is equal to \(E_{b}/N_{0}=9.8\) dB in the authentication system without the use of an error-correction code. When the BCH code is used, the signal-to-noise ratio is reduced to \(E_{b}/N_{0}=8.6\) dB. It is sufficient to have the signal-to-noise ratio be equal to \(E_{b}/N_{0}=7.6\) dB for an authentication system using an error-correction MPPC. Thus, it is obvious that the use of the developed algorithms of expansion and error correction in the MPPC allows for increasing the noise resistance of the authentication system using ZKAP based on the modular code. As a result  
Figure 4: Noise resistance curves for authentication systems: without the use of the error-correction code (**1**), using the BCH code (31, 11) (**2**) and using MPPC (30, 10) (**3**). of the MPPC application, there was a decrease in the probability of the first-order error, which contributes to improving the cybersecurity of the satellite IoV. To evaluate the effectiveness of the developed noise-resistant authentication protocol in countering spoofing-based cyberattacks, a prototype of the satellite authentication system was developed. Kintex UltraScale FPGA (xcku3p-frva676-1-e) was used to implement the prototype. The Xilinx Vivado-HLS 2018 CAD Xilinx Vivado-HLS was used to estimate the time cost required for authentication without considering the signal propagation time. The bit depth of the response signal coming from the SRSU is 120 bits. This signal has five parts: two digests and three responses. So, they are 24 bits each. Therefore, we will carry out a comparative analysis with the protocol [65], with the number \(S=20\),407,339 chosen as a module. 6-bit, 8-bit and 12-bit polynomials can be used to implement the MPPC. When implementing the protocol, only informational bases tuples will be selected for the MPPC. The possible choices are summarized in Table 2. The irreducible polynomials were taken from [74], which uses base-8 numbers for compact notation. Table 3 shows the time cost required to compute the SRSU's true and distorted digests. Table 4 summarizes the time the SRSU spends on generating the responses. Table 5 summarizes the time spent by the OBU to verify the response signal received from the SRSU. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bases Tuple Option** & \
\begin{tabular}{c} **Number of** \\ **Bases** \\ \end{tabular} & **DSP** & **FF** & **LUT** &
\begin{tabular}{c} **Computation** \\ **Time, ns** \\ \end{tabular} \\ \hline \(1\) & 4 & 9 & 92 & 698 & 43 \\ \(2\) & 3 & 9 & 117 & 663 & 57 \\ \(3\) & 2 & 9 & 185 & 623 & 64 \\ \(S=20\),407,339 & 1 & 10 & 202 & 583 & 107 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Time cost for generating the responses. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Option** & \(\mathbf{m_{1}}(x)\) & \(\mathbf{m_{2}}(x)\) & \(\mathbf{m_{3}}(x)\) & \(\mathbf{m_{4}}(x)\) \\ \hline \(1\) & 103 & 111 & 127 & 133 \\ \(2\) & 435 & 453 & 455 & \\ \(3\) & 15,647 & 16,533 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MPPC bases tuple options for ZKAP.