# An Ordered Universal Accumulator Based on a Hash Chain
## 5 Results
### Applications  
The hash chain accumulator, as a fundamental cryptographic tool, has application potential far beyond what is currently utilized, from data integrity verification to the membership witness to the core of blockchain technology. Its unique one-way and collision-resistant properties will play a crucial role in the digital realm.  
Hash chain accumulators have proven their efficiency and security across a wide range of internet applications. Our team's deployment, whether between traditional internet user terminals (MHC, message hash chain) and gateways or among different gateways, effectively addresses the stringent requirements for security, confidentiality, reliability, and traceability in communications between internal and external networks, as well as among disparate networks. Building upon this, the application of hash-chain-based software in Android apps, embedded development boards, and IoT routers establishes a chained trusted network architecture. This enables the creation of novel intrinsic network security technologies, including chained message transmission and trusted routing, effectively mitigating the inherent security vulnerabilities of existing IP network technology.  
Figure 4 illustrates our team's recent application of the hash chain accumulator in the Industrial Internet of Things (IIoT). By harnessing the immutable nature of the hash chain, we achieved effective integrity management for message parameters in data communication. We developed an IIoT authentication and communication scheme that integrates key security features, including privacy protection, data integrity, and traceability. This hash-chain-based secure communication scheme offers a lightweight and scalable alternative to traditional methods, such as precomputed digital signatures and MAC schemes.  
As shown in Figure 5, the hash-chain-based secure communication model establishes secure communication through a pre-generated keychain of secret keys, created in reverse  
Figure 4: Wide application range of hash chain accumulators.  
order using a one-way hash function, starting from a final secret key and securely distributing the initial key. Communication is structured into intervals, with each interval employing a specific key from the keychain. Within each interval, messages are authenticated using HMAC and the interval-specific secret key. Crucially, after each interval concludes, the corresponding secret key is disclosed. This delayed key disclosure, combined with the one-way hash function, ensures forward security, meaning that even if a key is compromised after disclosure, the security of past communication intervals remains unaffected, with only current and future intervals potentially at risk.  
The sender or receiver initiates a traceability request for a specific message \(\mathrm{m_{i}}\). For message \(\mathrm{m_{i}}\), the corresponding message element is computed: \(x_{i}=H(m_{i}||seq_{i})\). The sender or receiver, for message \(\mathrm{m_{i}}\), generates the corresponding message traceability evidence \(w_{i}\) = (\(w_{a}\),\(w_{b}\)), where \(w_{a}=[HAcc_{0},HAcc_{d},HAcc_{2d},\ldots,HAcc_{n}]\) and \(w_{b}=[x_{j,a},\ldots,x_{i-1},x_{i+1},\ldots,x_{(j+1),sd}]\), where d is the evidence node interval, and d = sqrt(n).  
The sender or receiver verifies the existence of message \(\mathrm{m_{i}}\) based on the VerHAcc algorithm within the traceability function and the message traceability evidence \(\mathrm{w_{i}}\). Traceable Success (Message Exists): if \(HAcc_{(j+1)*d}=H(HAC_{jnd}||\ldots||x_{i}||\ldots||x_{(j+1)*d})\), traceability is successful, indicating the message exists. Traceability Failure (Message Does Not Exist): If \(HAC_{(j+1)*d}\
eq H(HAC_{j*d}||\ldots||x_{i}||\ldots||x_{(j+1)*d})\), traceability fails, indicating the message does not exist.