# Autonomous Mutual Authentication Protocol in the Edge Networks
## 4 Analysis of Hardness of Autonomous Protocol  
**Theorem 1**: _If the multiple communicating parties as sender \(S\) and receiver\(R\) can successfully complete the authentication process, in such cases the validating receiver \(R\) always accept the sender \(S\) as valid._  
**Proof of Theorem.** According to the autonomous protocol process with reference to Section 3.3 Authentication Phase. If the sender \(S\) and receiver \(R\) are equipped with the authentication process with the secret of octet-group balanced tree, key generation, cryptographic algorithm, time-quarters, challenge-response within the multiple parties, then the challenge is initiated by the receiver \(R\) as  
\[R\ \to S:\ E_{PK_{S}}(K_{R,S}\parallel M_{8\times 8}\parallel\ \lambda_{R} \parallel PK_{R}\parallel ttl)\]Therefore, \(S\to R:\ E_{PK_{R}}\left(K_{S,R}\parallel\ \lambda_{S}\parallel T_{S_{2}}\parallel a_{c}\right)\) is confirmed to be valid when the receiver combines the temporal session key to be the final session key as \(R\ \to S:\ E_{PK_{S}}\left(``Authenticated\"\right)\). Ultimately, the successful completion confirms the authentication. \(\square\)  
**Theorem 2**.: _Considering the constructed possible solutions for the two challenges of octet-group balanced-tree transitions by the malicious user \(\hat{M}_{u}\), while assuming he is highly capable as receiver \(R\). When \(\hat{M}_{u}\) impersonated \(R\) and initiates the autonomous protocol process to approach \(S\) that he is real \(R\), in such case the probability of \(\hat{M}_{u}\) success is quite high._  
Proof of Theorem.: The search for a solution is intractable by the zero-knowledge proofs (ZKP) process for the transitions in octet-group balanced tree. Therefore, the possible solutions to this problem are equivalent to computing matrix multiplications by the number theory concepts. For challenge 1, \(\hat{M}_{u}\) doses a subtree to perform possible transitions during the attack. In every case of the challenge event, it is worth noting that the PRNG generates different parameters based on pseudo-random numbers, which are combined as \(\oplus\) with a time-stamp to generate session keys for all the autonomous edge devices. Thus, \(\hat{M}_{u}\) required for challenge 1 is to provide accurate tree nodes and its respective transition combination as the solution.  
\[R\ \to S:\ E_{PK_{S}}(K_{R,S}\parallel M_{8\times 8}\parallel\ \lambda_{R} \parallel PK_{R}\parallel ttl)\]  
For challenge 2, \(\hat{M}_{u}\) needs to achieve the octet-based transition related to time-stamp-based parameters. Multiple octet values are stored as a transition, which is combined as \(\oplus\) with the time-stamp as the final shared session key required to be calculated within a time limit.  
\[S\to R:\ E_{PK_{R}}\left(K_{S,R}\parallel\ \lambda_{S}\parallel T_{S_{2}} \parallel a_{c}\right)\]  
\(\square\)  
**Theorem 3**.: _Autonomous system authentication is a ZKP protocol._  
Proof of Theorem.: In the autonomous protocol process, the temporal session keys \(K_{R,S}\) and \(K_{S,R}\) exchange are encrypted by the public keys of both the parties that are part of the challenge-response scheme. Consecutively, the use of pseudo-random numbers, transitions values, octet-based balanced tree, and time-stamp combination makes the protocol process very hard to analyze and construct an accurate solution. Ultimately, it can be noted that \(\hat{M}_{u}\) is unable to devise a time-quarter-based solution and guess about any possible solution either for the challenge or ZKP. Henceforth, a strong claim for autonomous system authentication in the edge network is a ZKP protocol. The autonomous authentication achieves the ZKP process between multiple devices and can defend \(\hat{M}_{u}\)'s impersonation attack. \(\square\)