## 5 Blockchain-Based Supply Chain Security Solutions and Integration Models
### Data Privacy with Zero-Knowledge Proof (ZKP) and Confidential Transactions  
Blockchains are increasingly adopted in enterprise processes, such as trade finance, supply chain management, and contract execution. While companies prefer permissioned blockchains to protect commercial privacy, certain information must be transparently shared to ensure investor trust and traceability. In this context, striking a balance between commercial privacy and transparency is critical. To achieve this balance, transaction data can be protected through homomorphic encryption, while transparency can be maintained by employing zero-knowledge proofs. Thus, the validity of transactions can be verified without revealing sensitive information to unauthorized parties, and identities can be disclosed when necessary [86]. ZKP is a cryptographic technique that allows a prover to demonstrate the validity of a statement to a verifier without revealing any additional information. This mechanism ensures that the verifier can confirm the correctness of the provided information without accessing the underlying data itself [87]. ZKP protocols are grounded in three core mathematical properties: completeness, soundness, and zero-knowledge. These properties ensure that a party called the prover can convince another party, called the verifier of the truth of a statement, without revealing any information beyond the validity of the statement itself. While these definitions originate in theoretical cryptography, they translate into critical functionality for supply chain systems where privacy, trust, and fraud resistance are operational priorities [87, 88]. _Completeness_ ensures that a valid statement is always accepted. In other words, if the stated claim is true, the verifier must always accept the proof, as defined in Equation (1):  
\[\forall\in L,\ if\ P(x)\ is\ honest\Rightarrow V(x)=accept \tag{1}\]  
where x represents an instance from the language, \(L\); \(P(x)\) is the prover generating the proof; and \(V(x)\) is the verifier's decision. In supply chains, this guarantees that genuine data, such as IoT-recorded cold-chain temperature logs, will always be accepted by the verification system, supporting accurate compliance audits. _Soundness_ ensures that false claims are rejected with high probability. If the statement is false, the verifier must not accept fraudulent proof, as stated in Equation (2)  
\[\forall\
otin L,\ Probability[V(x)=accept]<\mathcal{E} \tag{2}\]  
where \(x\
otin L\) denotes an invalid instance, and \(\mathcal{E}\) is a negligible probability bound. This property is critical in preventing counterfeiters or malicious actors from introducing fraudulent data, such as fabricated certificates or falsified shipment origins. _Zero-knowledge_ ensures that no information is revealed beyond the validity of the claim. That is, the prover must not reveal any information other than proving that the statement is true, which is formalized in Equation (3):  
\[\exists S:S(x)\approx View_{v}(P(x)) \tag{3}\]  
where \(S(x)\) is a simulator's output, and \(View_{v}(P(x))\) denotes the verifier's view during the interaction with the honest prover, \(P\). This protects commercial secrets within the supply chain. For instance, a manufacturer can prove that a delivery occurred within the agreed pricing terms, without revealing the actual price or contract details. These foundational ZKP properties, though originating from theoretical computer science, address real-world challenges in supply chains. Table 6 presents the mathematical foundations of ZKP and maps each property to a direct implication in blockchain-enabled supply chain management. These properties serve as the foundational logic of zero-knowledge-based transaction verification in modern supply chains. Their applicability spans scenarios from pharmaceutical traceability to donor verification in humanitarian logistics. By enabling trust without disclosure, ZKP mechanisms enhance both privacy and operational integrity, which are increasingly vital under evolving systemic demands.