## 3 Development of an Imitation-Resistant Zero-Knowledge Authentication Protocol Based on an MPPC
Then, the combination is \(\widetilde{U}^{1}(x)=(28,\,21,\,23,\,9,\,15,\,18)\). Let us find the pairwise products (28). \[ \begin{array}{l}\left|\widetilde{U}_{1}^{1}(x)B_{1}(x)\right|_{M_{2}(x)}=( 11100,\,0,\,00110,\,11011,\,11000,\,10110)\\ =\big{(}x^{4}+x^{3}+x^{2},\,0,\,x^{2}+x,\,x^{4}+x^{3}+x+1,\,x^{4}+x^{3},\,x^{4 }+x^{2}+x\big{)}.\\ \end{array}  \tag{66}\]  
\[ \begin{array}{l}\left|U_{2}^{1}(x)B_{2}(x)\right|_{M_{2}(x)}=(0,\,10101,\,1 000,\,1010,\,00101,\,10111)=\\ =\big{(}0,\,x^{4}+x^{2}+1,\,x^{4}+x^{3},\,x^{4}+x^{2}+x,\,x^{2}+1,\,x^{4}+x^{ 2}+x+1\big{)}.\\ \end{array}  \tag{67}\]  
Let us compute the error syndrome using (31):  
\[\lambda_{1}(x)=U_{3}(x)+\left|\sum_{j=1}^{2}\gamma_{3}^{j}(x)\right|_{m_{3}(x )}=10111+11110=01001. \tag{68}\]  
\[\lambda_{2}(x)=U_{4}(x)+\left|\sum_{j=1}^{2}\gamma_{4}^{j}(x)\right|_{m_{4}(x )}=01001+01101=00101. \tag{69}\]  
\[\lambda_{3}(x)=U_{5}(x)+\left|\sum_{j=1}^{2}\gamma_{5}^{j}(x)\right|_{m_{5}(x )}=11101+01111=10010. \tag{70}\]  
\[\lambda_{4}(x)=U_{6}(x)+\left|\sum_{j=1}^{2}\gamma_{6}^{j}(x)\right|_{m_{6}(x )}=10010+00001=10011. \tag{71}\]  
The error syndrome is not zero. This means that the MPPC combination contains an error. The error syndrome is fed to the input of the memory block, where the error vectors for two informational residues are stored. Since only the informational residues are involved in checking the response signal, we will correct only them. In our example, the error vector is equal to \(\overline{z}=(00010,\,00000)\). To correct the error, we need to add the informational residues of the distorted combination modulo two with the error vector. We obtain \(U^{1}(x)=\widetilde{U}^{1}(x)+\overline{\epsilon}(x)=(111\bar{0}0,\,10101)+(00 010,\,0000)=(11110,\,10101)\). The error in the first informational residue has been corrected. The verification of the SRSU's signal is as follows:  
\[ \begin{array}{l}OBU:A_{1}(1)=\left|\big{(}x^{4}+x^{3}+x^{2}+x\big{)}^{8}x^{ 26}x^{29}x^{13}\right|_{m_{1}(x)}=x^{4}+x^{3}+x^{2}+1=11101=29_{10}.\\ OBU:A_{2}(1)=\left|\big{(}x^{4}+x^{2}+1\big{)}^{4}x^{18}x^{12}x^{23}\right|_{m_ {2}(x)}=x^{4}+x+1=10011=19_{10}.\\ \end{array}  \tag{72}\]  
Since \((A_{1}(1),A_{2}(1))=(\hat{U}_{1}(1),\hat{U}_{2}(1))=(29,19)\), the SRSU's status is \"friend\" and it is granted a communication session with the OBU. In the above example, the ability of the authentication protocol to effectively resist interference has been demonstrated. As a result of the developed noise-resistant protocol, the probability of the first-order error, when a legitimate SRSU is not allowed to start the communication session, will be reduced. To evaluate the effective impact of the redundant MPPC on the noise resistance of the SRSU authentication system, the modeling of the response signal passing through a channel with additive white Gaussian noise (AWGN) for different values of the signal-to-noise (\(E_{b}/N_{0}\)) ratio was conducted. A total of 1000 experiments were performed. A total of 10,000 response signals were transmitted in each experiment. The confidence interval is 95%. The results of noise resistance studies of authentication systems are shown in Figure 4. The developed noise-resistant MPPC was compared with a system that does not use a noise-resistant code, as well as with a burst error-correction BCH code. The developed error-correction MPPC (30, 10) was compared with the BCH code (31, 11). These codes allow for correcting an error burst containing up to four bits. In the error-correction MPPC code, \(k=2\) information bases are used. These are the polynomials \(p_{1}(x)=x^{5}+x^{4}+x^{3}+x^{2}+1\) and \(p_{2}(x)=x^{5}+x^{4}+x^{3}+x+1\). The redundant bases are \(p_{3}(x)=x^{5}+x^{4}+x^{2}+x+1\), \(p_{4}(x)=x^{5}+x^{3}+x^{2}+x+1\), \(p_{5}(x)=x^{5}+x^{3}+1\) and \(p_{6}(x)=x^{5}+x^{2}+1\). This code can correct twofold errors, i.e., the distortion of two residues in the code combination. The MPPC decoder presented in the paper uses a hard decoding scheme.