### Contribution  
The contributions/objectives to the hash-chain fog/edge can be given as follows:  
Recently, several mutual authentication systems have used the blockchain distributed ledger, which helps to authenticate devices within the network. Subsequently, solutions have been developed as well, but they are incapable of operating without a third-party system. Thus, there is a need to design a system that can operate without a third-party system. To secure the device by authentication, identities are cross-checked from the distributed database system, which is then mutually authenticated by a novel hash-chain protocol supported by a challenge-response model of zero-knowledge proofs. Eventually, the hash-chain fog/edge objectives can be presented as follows:  
* Solving the centralized identity key management problem for authentication: Identity key management is a crucial factor for the authentication system. Authenticating a particular device needs to be first verified by the centralized server. Therefore, all the devices/systems within the fog need to be dependent on the centralized server for authentication. However, the fog also needs to update itself to a centralized server for identity key management, which may face a single-point failure. The hash-chain fog/edge provides distributed identity key management with the help of a distributed database system (DDS). The DDS is a popular model for identity key management;
* Novel algorithm for mutual authentication within the fog/edge: Multiple devices/systems need to be uniquely authenticated based on transitions in the challenge-response model for the hash-chain flow. The proposed model is adaptable and capable of uniquely identifying and authenticating devices/systems by the dynamic challenge-response model. The selection of the transition matrix for the challenge model within the sub-branch of the PRN-based random number tree is performed for every device/system across multiple fogs using several ideas from interactive proof systems;
* Proving dynamic authentication by using zero-knowledge proofs (ZKPs): All the devices/systems within the multiple fogs need to solve the challenge-response model for authentication. The three basic properties of ZKP include completeness, in which the required operations need to be achieved within the specific conditions, soundness, where no cheating devices/systems can succeed in convincing an honest authenticating server, and zero-knowledge, in which the devices/system can never gain complete knowledge about the transition/working secret. These properties help us authenticate the devices/systems, as they possess the part of the knowledge to solve the challenge, which is then verified by the authenticating server;
* Usage of minimum infrastructure and less calculation on the IoT device: The use of minimum infrastructure for authentication within the fog/edge helps us eliminate theticket-granting server (TGS) and the separate authentication server (AS), other than the fog server and service server.  
Thus, no new system need to be installed within this model, which is a virtual network of fog/edge computing. All the authentication is performed uniquely as we use a pseudo-random number generator (PRNG), which helps us make fewer calculations on resource-constrained IoT devices that are configured with low memory capacity and processing capability.