## 3 Preliminaries
### Blockchain  
Blockchain is a distributed ledger composed of a sequence of blocks \(B_{0},B_{1},\ldots,B_{n}\), where each block contains a set of validated transactions. A blockchain system can be formally modeled as a state-based transition system \(M=(S,I,\delta,s_{0})\), where the following are true:  
* \(S\) is the set of global states. The global state refers to the complete and shared snapshot of all relevant data that defines the current status of the blockchain network. This includes information such as account balances, smart contract storage, and other protocol-defined data. * \(I\) is the set of valid inputs (e.g., transactions, blocks). * \(\delta:S\times I\to S\) is the deterministic state transition function. * \(s_{0}\in S\) is the initial state. The evolution of the blockchain system is described by the following transition sequence:  
\[s_{0}\xrightarrow{i_{0}}s_{1}\xrightarrow{i_{1}}s_{2}...\xrightarrow{i_{n-1}} s_{n}\]  
Each state \(s_{k}\) may represent the current ledger view, set of valid transactions, and configuration of participating nodes. Consensus protocols determine how a new input \(i_{k}\) is selected and verified before applying \(\delta\). A block B can be defined generically as  
\[B=(header,T)\]  
where \(T=\{tx_{1},tx_{2},...,tx_{m}\}\) is the set of transactions and the _header_ contains metadata such as the previous block hash \(h_{prev}\) timestamp \(t\) and consensus-specific proof \(\pi\). The validity of a block is determined by a predicate:\[Valid(B,s)\Leftrightarrow Verify(T,s)\wedge ConsensusCheck(\pi,s)\]  
This model accommodates various consensus mechanisms. For example, in PoW, \(ConsensusCheck(\pi,s)\) verifies that the proof satisfies \(h(B)<D\), where \(D\) is the difficulty target. In PBFT, it verifies a quorum of signed messages from validator nodes. Blockchains can be broadly classified into public, private, consortium, and hybrid types. This classification is based on factors such as access permissions, governance models, and intended applications. Public blockchains are open and permissionless. Anyone can join the network, read data, submit transactions, and participate in consensus. These networks are fully decentralized and trustless. Public blockchains are well-suited for open systems where transparency, immutability, and user autonomy are essential. Private blockchains, also known as permissioned blockchains, restrict access to a single organization or a specific group. Only authorized users can validate transactions, access records, or operate nodes. These blockchains are often used in enterprise environments. They enhance data integrity, enable traceability, and support process automation, while keeping sensitive information private. Consortium blockchains sit between public and private models. They are jointly managed by a group of predefined organizations, creating a decentralized but controlled network. Each member operates a node and takes part in the consensus process. This model is effective for sectors like finance, supply chain, and healthcare, where multiple parties need to share data without depending on a single authority. Hybrid blockchains combine features of both public and private systems. They support selective transparency--some data can be public, while other parts remain private or permissioned. This approach offers both flexibility and scalability. It is suitable for use cases that require public verification along with controlled access, such as government services, enterprise systems, and IoT platforms. In the context of the IIoT, consortium blockchains offer a robust solution for device authentication by enabling a decentralized yet collaboratively governed registry of devices. This approach reduces dependence on a single centralized Certificate Authority while maintaining control within a trusted group of stakeholders. Consortium blockchains enhance resistance to spoofing and unauthorized access and support secure, auditable interactions among industrial entities.