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Authentication is essential for the prevention of various types of attacks in fog/edge computing. Therefore, a novel mode-based hash chain for secure mutual authentication is necessary to address the Internet of Things (IoT) devices' vulnerability, as there have been several years of growing concerns regarding their security. Therefore, a novel model is designed that is stronger and effective against any kind of unauthorized attack, as IoT devices' vulnerability is on the rise due to the mass production of IoT devices (embedded processors, camera, sensors, etc.), which ignore the basic security requirements (passwords, secure communication), making them vulnerable and easily accessible. Furthermore, crackable passwords indicate that the security measures taken are insufficient. As per the recent studies, several applications regarding its requirements are the IoT distributed denial of service attack (IDDOS), micro-cloud, secure university, Secure Industry 4.0, secure government, secure country, etc. The problem statement is formulated as the "design and implementation of dynamically interconnecting fog servers and edge devices using the mode-based hash chain for secure mutual authentication protocol", which is stated to be an NP-complete problem. The hash-chain fog/edge implementation using timestamps, mode-based hash chaining, the zero-knowledge proof property, a distributed database/blockchain, and cryptography techniques can be utilized to establish the connection of smart devices in large numbers securely. The hash-chain fog/edge uses blockchain for identity management only, which is used to store the public keys in distributed ledger form, and all these keys are immutable. In addition, it has no overhead and is highly secure as it performs fewer calculations and requires minimum infrastructure. Therefore, we designed the hash-chain fog/edge (HCFE) protocol, which provides a novel mutual authentication scheme for effective session key agreement (using ZKP properties) with secure protocol communications. The experiment outcomes proved that the hash-chain fog/edge is more efficient at interconnecting various devices and competed favorably in the benchmark comparison. [ABSTRACT FROM AUTHOR]