## 3 Development of an Imitation-Resistant Zero-Knowledge Authentication Protocol Based on an MPPC
\begin{array}{l}\lambda_{1}(x)=U_{k+1}(x)+\left|\sum\limits_{j=1}^{k} \gamma_{k+1}^{j}(x)\right|_{m_{k+1}(x)},\\ \vdots\\ \lambda_{r}(x)=U_{k+r}(x)+\left|\sum\limits_{j=1}^{k}\gamma_{k+r}^{j}(x) \right|_{m_{k+r}(x)}.\end{array} \right. \tag{31}\]  
The analysis of the expression shows that it is sufficient to use only modulo-two multiple-input adders to calculate the error syndrome. We will perform a comparative performance analysis of the developed algorithm for error search and correction with previously known algorithms using Kintex UltraScale FPGA (xcku3p-ffva676-1-e) and Xilinx Vivado-HLS 2018 CAD Xilinx Vivado-HLS. The number of informational bases is \(k=4\). The number of control bases is \(r=2\). The irreducible polynomials are represented in an octal number system. In this case, \(m_{1}(x)=103_{8}=1000011=x^{6}+x+1\). The analysis of Table 1 shows that the developed algorithm allows fo 3.5 times faster error correction than algorithm [71], 3.7 times faster error correction than algorithm [72] and 36.2 times faster error correction than algorithm [50]. Thus, the simulation results showed that the developed algorithm requires less time to locate and correct errors in MPPC. In order to implement a noise-resistant authentication protocol, it is necessary to introduce redundancy into the code combination by expanding the residue tuple before transmitting the signal. This result is obtained by expanding the set of bases. In this case, it is necessary to calculate the residue \(U_{k+1}(x)\) on the redundant base \(m_{k+1}(x)\) such that  
\[U_{k+1}(x)\equiv U(x)\text{mod}m_{k+1}(x) \tag{32}\]  
In this case, it is necessary to use the values of informational residues \(U(x)=(\ U_{1}(x)\), \(U_{2}(x)\),..., \(U_{k}(x))\) without conversion into a positional code. Obviously, the algorithm for expanding the set of bases in the MPPC should coincide with the algorithm used for error detection and correction. Let us use the developed algorithm, which uses orthogonal bases with broken orthogonality on redundant bases. We will use expressions (29) and (30) to extend the set of bases. Then, the values of the redundant residues are  
\[\left\{ \begin{array}{l}U_{k+1}(x)=\sum\limits_{i=1}^{k}\gamma_{k+1}^{j}(x) \text{mod}m_{k+1}(x),\\ \vdots\\ U_{k+r}(x)=\sum\limits_{i=1}^{k}\gamma_{k+r}^{i}(x)\text{mod}m_{k+r}(x).\end{array} \right. \tag{33}\]  
The analysis of expression (33) shows that control residues can be computed in parallel for \(r\) redundant bases, using only multiple output adders modulo two for this purpose. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bases** & **Algorithm [50]** & **Algorithm [71]** & **Algorithm [72]** & **Developed** \\ \hline Informational: & & & & **Protocol** \\
103, 127, 133, 141 & & & & & \\ Redundant: 141, & & & & & \\
143 & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time cost of error correction. The error burst correction (31) and residues expansion (33) algorithms will be used in the developed authentication protocol based on MPPC. Analysis of the Effectiveness of Ensuring Cybersecurity of Satellite IoV with the Developed Noise-Resistant ZKAP  
Let two informational bases \(m_{1}(x)=x^{5}+x^{3}+1,\ m_{2}(x)=x^{5}+x^{2}+1\) be chosen to implement the developed noise-resistant ZKAP for a satellite VANET system. The range of allowed combinations in MPPC is \(M_{2}(x)=x^{10}+x^{8}+x^{7}+x^{5}+x^{3}+x^{2}+1\), so the secret parameters should be less than \(2^{10}\). The polynomials \(m_{3}(x)=x^{5}+x^{3}+x^{2}+x+1\), \(m_{4}(x)=x^{5}+x^{4}+x^{2}+x+1\), \(m_{5}(x)=x^{5}+x^{4}+x^{3}+x+1\) and \(m_{6}(x)=x^{5}+x^{4}+x^{3}+x^{2}+1\) are chosen as redundant bases. The resulting code (30, 10) can correct errors that can occur simultaneously in two residues of the combination. Let us consider the implementation of ZKAP under the influence of interference in the communication channel.