# Adaptive Authentication Protocol Based on Zero-Knowledge Proof
## 6 Analysis of the Results of the Conducted Research  
The VANET model was developed to analyze the parameters of the developed adaptive authentication protocol. NS-2 was chosen as the simulation system, in which code modification is allowed. The VANET model consists of 10 \(RSU\)s. The maximal number of \(OBU\)s per \(RSU\) is 100. Each \(OBU\) and \(RSU\) uses the developed adaptive authentication protocol. Communication channel is a discrete communication channel without interference. The radius of the interaction zone of the \(OBU\) and \(RSU\) is 1 km. \(RSU\)s are 2 km apart to provide direct visibility. The data transfer rate between \(OBU\) and \(RSU\) is 1 Mbit/s. The time-to-live (TTL) of VANET network is 1. Minimal packet size is 50 bytes. Maximal packet size is 200 bytes. The cryptographic strength of the developed adaptive authentication protocol will be determined by the bit depth of the signals \(S_{OBU}(i)\), \(S_{RSU}(i)\) coming from the prover. We will use the probability of selecting the prover signal to assess the cryptographic strength of the adaptive protocol:  
\[P=\frac{1}{M_{j}2^{\log_{2}D}}, \tag{34}\]  
where \(M_{j}\) is the number of parts in signals \(S_{OBU}(i)\), \(S_{RSU}(i)\). Consider the use of a 32-bit modulus in the developed authentication protocol. The number of parts of \(S_{OBU}(i)\), \(S_{RSU}(i)\) is \(M_{j}=4\) with a maximal level of confidentiality. Then, the probability of selecting the prover signal is \(P_{\max}^{(32)}=5.82\cdot 10^{-11}\). If the level of confidentiality is reduced to a minimum, the number of parts of \(S_{OBU}(i)\), \(S_{RSU}(i)\) will be reduced to \(M_{j}=3\). In this case, the probability of selecting the prover signal is \(P_{\min}^{(32)}=7.76\cdot 10^{-11}\). Thus, the transition from maximal level of confidentiality to minimal one leads to a decrease in the cryptographic strength of the protocol by 1.33 times. However, the authentication protocol using the 32-bit modulus \(D\) does not provide a high level of confidentiality. This is confirmed by a software implementation that allows us to determine the password's resistance to brute force [46]. If a 56-bit password is used, it will be cracked after 2 h. If the password length is 64 bits, then the time interval required to crack the password increases to 2 days. Therefore, it is obvious that the bit depth of modulus \(D\) should not be less than 64 bits. Therefore, when using a 64-bit modulus and maximal level of confidentiality for the developed protocol, the probability of selecting the prover signal is \(P_{\mathrm{max}}^{(64)}=1.35\cdot 10^{-20}\). When the level of confidentiality is reduced to a minimum, the probability of selecting the prover signal is \(P_{\mathrm{min}}^{(64)}=1.81\cdot 10^{-20}\). Increasing the size of modulus \(D\) to 64 bits allowed for a confidentiality level that was more than eight orders of magnitude higher compared to the 32-bit authentication protocol. The FPGA Artix-7 xc7a12 ticsg325-1L was used to evaluate the effectiveness of the developed adaptive authentication protocol. The clock frequency was 100 MHz. Testing was conducted on the Vivado HLS 2019 platform. The bit depth of the modulus \(D\) used in the protocol was 32. The multiplicative operation was based on a binary algorithm, which was used to exponentiate an integer modulo \(D\). The following parameters were selected to determine the maximal possible time spent on authentication: modulus \(D\) = 4,294,967,291, an exponent ranging from 20 to 30. With the use of a binary algorithm to exponentiate an integer, the calculation time of the true digest was \(T_{1}\) = 14,200 ns and the calculation time of the noisy digest was \(T_{2}\) = 15,650 ns. The transmission time of the question was \(T_{3}\) = 32,000 ns. The calculation time of the answer to the question was \(T_{4}\) = 400 ns. The transmission time of the answer to the question was \(T_{5}\) = 96,000 ns. The answer check time was \(T_{6}\) = 28,530 ns.