## 5 Security Analysis  
The security analysis of the proposed authentication framework is performed using the informal security analysis methods; the resilience of the proposed authentication framework against various attacks has been discussed, as in [28]. Proposition 1: _Resilience against replay attack._  
Proof: In the proposed authentication scheme, all communicating parties (IoT devices and Blockchain) randomly generate keys and variables, such as _sdij_, _xij_, _sbij_, _degij_, and then compute variables, such as _Tdevij_, _Tblockij_, _Tdevblockij_, and _Mij_, each time an authentication request is made. Each communication and participating variable is transferred with a timestamp. Hence, a fresh generation of variables and the usage of timestamps for each authentication request support the proposed framework against the replay attack. Proposition 2: _Resilience against man-in-the-middle attack._  
Proof: The use of Chebyshev polynomials in the authentication mechanism makes it resilient against man-in-the-middle attacks. The Blockchain and each participating IoT device share a piece of secret (_degij_) in one-to-one communication. This communication is secured and no IoT device can know the piece of the secret of another IoT device. Another piece of secret (_chebyshev_val_ is securely shared between the Blockchain and individual IoT devices in separate communications. The Blockchain will approve authentication only if all IoT devices compute their corresponding _chebyshev_val_ using their secret. If someone successfully intercepts one communication he/she cannot recreate another valid login request method. He/she cannot even use the intercepted communication to recreate another authentication request for the same IoT device as Blockchain will securely share the next secret (_degij_) using _1 Dpurf_ of the listed IoT device only. Hence, obtaining (_degij_) of one IoT device will not help a man-in-the-middle attacker to successfully recreate another authentication request. Proposition 3: _Resilience against offline guessing attacks._  
Proof: In the proposed authentication framework, Blockchain initiates the process and _sdij_ and _xij_ are randomly generated for each authentication request; offline guessing attack will not help an attacker to use stored _sdij_ and _xij_ for further authentication cracking. Retrieval of _sdij_ and _xij_ from one IoT device cannot help cracking authentication for other devices as well as further authentication for the same device; hence, we can say that the proposed framework is resilient against offline guessing attacks. Proposition 4: _Resilience against device impersonation attacks and lost/stolen IoT device attacks._  
Proof: In the group formation phase, the trusted authority uses the metadata _1 Dpurf_, along with other information, to create a _blockCert_dev_. This _1 Dpurf_ is a physical unclonable function that is unique to each device, and any communication can be verified to be sent only by that device. Since _blockCert_dev_ is stored on both the Blockchain and IoT devices, even if someone changes the IoT device's information, it can be verified from the certificate fetched from the Blockchain. This makes it difficult for anyone to impersonate an IoT device. Moreover, because the digital certificate of an IoT device can be verified from the Blockchain, it is easy to block lost/stolen devices from participating in future authentication requests. An invalidated certificate is stored on the Blockchain for the lost/stolen device, and since the Blockchain is searched from the latest-to-oldest fashion, the updated certificate will be fetched first, which will show that the device is no longer valid for that group. Hence, we can say that the proposed authentication framework is resilient against impersonation and lost/stolen device attacks. **Proposition 5**: _Resilience against the ephemeral secret leakage (ESL) attack._  
The security against ESL attacks depends on the long-term secret and temporal secret.