# An Ordered Universal Accumulator Based on a Hash Chain
## 5 Results
### Experimental Analysis  
We tested the efficiency of the HC model, the RSA model, and the Merkle tree. The hardware configuration included an Ubuntu 20.04 operating system, a CPU with 2.9 GHz, an RTX 2060, and a memory with 16 GB. The models were written in the Python computing language running on Pycharm CE version 2023.1.4. Each experiment for a given set size was run 30 times for each accumulator type to ensure the stability and reliability of the timing measurements. In the experimental evaluation, the main measurement objectives included membership proof generation time cost and verification time cost. Figure 2 displays the results of the evidence generation efficiency of the three models (across all three accumulators, evidence generation times did not significantly vary across set sizes, as indicated by non-significant ANOVAs (\(p>0.10\) for all tests)). The efficiency with which the RSA model generated evidence is linear, while the HC model and Merkle tree are virtually unaffected by the set size. The RSA model takes about 10-40 times longer to generate evidence \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & **Security** & \multirow{2}{*}{**Scheme**} & \multirow{2}{*}{**S\(\text{ND}\)**} & \multirow{2}{*}{**U**} & \multirow{2}{*}{**Lmt**} & \multirow{2}{*}{**Trapdoor**} & \multirow{2}{*}{\(|\mathbf{pp}|\)} & \multirow{2}{*}{\(|\mathbf{w}|\)} & \multirow{2}{*}{\(|\mathbf{u}|\)} \\  & **Assumption** & & & & & & & & \\ \hline \multirow{3}{*}{RSA-based} & \multirow{3}{*}{S-RSA} & BM [11] & S & - & - & \(\surd\) & \(O(1)\) & \(O(1)\) & - \\  & & CL [12] & D & - & - & \(\surd\) & \(O(1)\) & \(O(1)\) & - \\  & & BBF [24] & D & \(\surd\) & - & \(\surd\) & \(O(1)\) & \(O(1)\) & \(O(1)\) \\ \hline \multirow{2}{*}{Bilinear-based} & \multirow{2}{*}{\(q\)-SDH} & DT [32] & D & \(\surd\) & \(\surd\) & \(\surd\) & \(O(q)\) & \(O(1)\) & \(O(1)\) \\  & & CKS [17] & D & - & \(\surd\) & - & \(O(q)\) & \(O(1)\) & - \\ \hline \multirow{2}{*}{Hash-based} & \multirow{2}{*}{CRH} & CHKO [20] & D & \(\surd\) & - & - & \(O(1)\) & \(O(log\text{N})\) & \(O(log\text{N})\) \\  & & HC & D & \(\surd\) & - & - & \(O(1)\) & \(O\Big{(}\sqrt{N}\Big{)}\) & \(O\Big{(}\sqrt{N}\Big{)}\) \\ \hline \hline \end{tabular} Note. \(\surd\) indicates conformity to the function or nature, and - indicates that it does not.  
\end{table} 
Table 2: Efficiency analysis. than the HC model and the Merkle tree. The size of the set has less impact on the efficiency of evidence generation and the validation of the HC model. The operation speed of the hash algorithm is higher than that of the RSA algorithm, so the efficiency of the HC model in actual operation is much higher than that of the RSA model. RSA model verification takes about 4 times longer than the HC model and 200 times longer than the Merkle tree. RSA accumulators rely on multiplication and modulo operations on large integers, which are computationally more expensive. In contrast, hash chain accumulators use hash functions for authentication, and commonly used hash functions (SHA-256) have efficient implementations on edge devices with less computational overhead for the authentication process. While RSA and bilinear mapping accumulators may require less storage space than hash-based accumulators, they introduce communication complexity during updates, as the evidence holder must retrieve a new witness from the accumulator manager. As shown in Figure 3, HC accumulators exhibit significantly lower storage costs compared to Merkle trees. The HC model requires only a sequential chain of hash values, minimizing storage overhead. In contrast, Merkle trees store both leaf and internal nodes, resulting in higher storage requirements. However, HC accumulators have a key limitation: deleting an element necessitates recalculating all subsequent hashes, with a time complexity of O(N). Merkle trees, by comparison, update only O(logN), making them better suited for dynamic datasets with frequent deletions.