TL;DR
- Mathematical modeling of L3 zkRollup architecture’s gas efficiency shows a theoretical gas cost reduction of over 99.8%.
- This efficiency stems from the combined effects of layered rollup structure, off-chain data availability (Validium), and the zkVerify verification network.
- While various constraints may reduce efficiency in actual implementation, significant improvements over existing solutions are still expected when considering the characteristics of complex state synchronization tasks.
- This analysis is based on theoretical models and results may vary with actual implementation and testing.
Introduction
One of the key challenges in blockchain technology is optimizing transaction processing costs, particularly gas costs on the Ethereum network. This extends beyond simple cost reduction. Gas efficiency is an essential prerequisite for effectively handling complex operations such as state synchronization, which determines the economic feasibility of innovative applications like Real World Asset (RWA) tokenization.
Oraclizer’s L3 zkRollup architecture integrates several technological innovations to maximize gas efficiency. In this article, we analyze how these technical innovations lead to substantial gas cost savings through mathematical modeling.
Unlike previous research, we comprehensively analyze the combined effects of L3 layering, the Validium approach, and the zkVerify network, considering the specificity of state synchronization operations. This analysis mathematically demonstrates Oraclizer’s technical advantages and presents both the theoretical upper limits of expected performance improvements and realistic expectations in actual implementation.
Efficiency Modeling Framework
To analyze the gas efficiency of the Oraclizer architecture, we will build a mathematical modeling framework and quantify the degree to which each technological element contributes to overall efficiency.
Efficiency Model of Layered Rollup Structure
To understand the efficiency of L3 zkRollup architecture, we must first model the transaction processing costs at each layer: L1 (Ethereum mainnet), L2 (Base), and L3 (Oraclizer). The transaction processing cost at each layer can be formulated as follows:
$C_i(n) = B_i + T_i \cdot n + P_i(n)$
Where:
- $C_i(n)$: Cost of processing $n$ transactions at layer $i$
- $B_i$: Basic batch overhead cost
- $T_i$: Individual transaction cost
- $P_i(n)$: Proof generation and verification cost (a function of the number of transactions)
The proof cost function $P_i(n)$ can be modeled as:
$P_i(n) = \alpha_i \cdot n^{\beta_i} + \gamma_i$
Where:
- $\alpha_i$: Proof complexity coefficient
- $\beta_i$: Scaling exponent ($0 < \beta_i \leq 1$)
- $\gamma_i$: Fixed verification overhead
In zkRollup systems, typically $\beta_i \approx 0.5$, meaning the proof size is proportional to the square root of the number of transactions.
Calculating Efficiency Gains Between Layers
The efficiency gain of L2 relative to L1 is calculated as:
$E_{L2/L1}(n) = \frac{C_1(n)}{C_1(1) + C_2(n)}$
Where the denominator $C_1(1)$ is the cost of a single transaction to submit the L2 batch result to L1.
Similarly, the efficiency gain of L3 relative to L2 is:
$E_{L3/L2}(n) = \frac{C_2(n)}{C_2(1) + C_3(n)}$
Therefore, the total efficiency gain of L3 relative to L1 is:
$E_{L3/L1}(n, m) = E_{L2/L1}(m) \cdot E_{L3/L2}(n)$
Where $m$ is the L2 batch size and $n$ is the L3 batch size.
Validium Data Availability Model
To model the effect of the off-chain data availability (Validium) approach, we introduce the data cost function:
$D_i(n) = \delta_i \cdot n \cdot s$
Where:
- $D_i(n)$: Data cost of $n$ transactions
- $\delta_i$: Unit data cost coefficient
- $s$: Average transaction size
In the Validium approach, the on-chain data cost is replaced by:
$D_{validium}(n) = \delta_i \cdot h$
Where $h$ is the fixed size of the data hash.
Therefore, the efficiency gain from Validium is:
$E_{validium}(n) = \frac{D_i(n)}{D_{validium}(n)} = \frac{\delta_i \cdot n \cdot s}{\delta_i \cdot h} = \frac{n \cdot s}{h}$
As can be seen in this formula, the efficiency gain from Validium increases proportionally with the number of transactions $n$. That is, the more transactions processed, the greater the benefits of off-chain data availability.
zkVerify Network Optimization Model
The zkVerify network provides additional efficiency improvements by optimizing the proof generation and verification process. To model this, we modify the proof generation cost function as follows:
$P_{zkVerify}(n) = \frac{P_i(n)}{\eta \cdot (1 + \log(k))}$
Where:
- $\eta$: Hardware optimization coefficient ($\eta > 1$)
- $k$: Number of parallel proof generators
- The $\log(k)$ term reflects the logarithmic scale improvement in proof generation time due to parallelization
In this model, the efficiency improvement by the zkVerify network is manifested as a combined effect of hardware optimization and parallel processing.
State Synchronization Characteristics and Efficiency Impact
State synchronization, Oraclizer’s core function, requires more complex calculations than general transaction processing. Let’s analyze how this complexity affects the efficiency model.
State Synchronization Complexity Model
The complexity of state synchronization tasks varies depending on the number of state changes to be synchronized and the interdependencies between them. This can be modeled by the following function:
$S(n, d) = \xi \cdot n \cdot (1 + \log(d))$
Where:
- $S(n, d)$: Cost of state synchronization with $n$ state changes and maximum dependency depth $d$
- $\xi$: State verification complexity coefficient
- The $\log(d)$ term reflects the increase in verification complexity with the depth of the dependency graph
This model indicates that verification complexity increases logarithmically as the dependency graph between state changes deepens. This is particularly important when synchronizing the state of complex real-world assets such as financial products.
Impact of State Synchronization on Efficiency
The impact of state synchronization tasks on overall efficiency can be modeled as:
$E_{state\_sync} = \frac{T_i \cdot n}{T_i \cdot n + S(n, d)}$
This formula shows the extent to which state synchronization costs, added to total transaction costs, reduce efficiency. This effect is more pronounced in large-scale state synchronization tasks with complex dependencies.
Integrated Efficiency Model
Now we can integrate all the previously explained factors to express the total efficiency gain of the L3 zkRollup architecture:
$E_{total}(n, m, d) = E_{L3/L1}(n, m) \cdot E_{validium}(n) \cdot \frac{P_i(n)}{P_{zkVerify}(n)} \cdot E_{state\_sync}$
When expressed as a percentage, the final efficiency improvement is:
$\text{Efficiency improvement} = \left(1 – \frac{1}{E_{total}}\right) \times 100\%$
This integrated model shows how the various technological innovations of the Oraclizer architecture interact to determine overall gas efficiency.
Parameter Estimation and Theoretical Efficiency Analysis
Based on actual Ethereum network data and the performance of existing L2 solutions, we will estimate the parameters of the model and calculate the theoretical efficiency.
Parameter Estimation
The table below shows the estimated values for the model parameters:
| Parameter | Description | Estimated Value |
|---|---|---|
| $B_1$ | L1 batch overhead | 21,000 gas |
| $T_1$ | L1 transaction cost | 65,000 gas |
| $B_2$ | L2 batch overhead | 5,000 gas |
| $T_2$ | L2 transaction cost | 2,500 gas |
| $\alpha_2, \beta_2$ | L2 proof parameters | 10,000, 0.5 |
| $B_3$ | L3 batch overhead | 1,200 gas |
| $T_3$ | L3 transaction cost | 500 gas |
| $\alpha_3, \beta_3$ | L3 proof parameters | 3,000, 0.5 |
| $s$ | Average transaction size | 150 bytes |
| $h$ | Hash size | 32 bytes |
| $\eta$ | Hardware optimization coefficient | 2.5 |
| $k$ | Number of parallel proof generators | 8 |
| $\xi$ | State verification complexity | 1,500 gas |
| $d$ | State dependency depth | 4 |
Theoretical Efficiency Calculation
Using the above parameters, let’s calculate the efficiency gain of each technological element at typical batch sizes ($n = 1000, m = 100$):
- L3 layering effect:
- $E_{L2/L1}(100) \approx 28.4$ (96.5% gas reduction)
- $E_{L3/L2}(1000) \approx 9.7$ (89.7% gas reduction)
- $E_{L3/L1}(1000, 100) \approx 275.5$ (99.6% gas reduction)
- Validium effect:
- $E_{validium}(1000) \approx 4.7$ (78.7% gas reduction)
- zkVerify effect:
- zkVerify improvement factor $\approx 2.8$ (64.3% gas reduction)
- State synchronization impact:
- $E_{state\_sync} \approx 0.63$ (37% efficiency reduction)
Integrating these individual effects gives us the following theoretical efficiency gain:
$E_{total} \approx 275.5 \times 4.7 \times 2.8 \times 0.63 \approx 2,281$
This means a theoretical efficiency improvement of about 2,281 times compared to L1, which corresponds to a gas cost reduction of approximately 99.956%.
Realistic Constraints and Actual Expectations
The theoretical model assumes ideal conditions, but in actual implementation, various constraints reduce efficiency. Let’s estimate the actual expectations considering these factors.
Implementation Overhead
In actual implementation, additional safeguards and overhead occur compared to the ideal model:
$E_{actual} = E_{theoretical} \cdot (1 – \omega_{impl})$
Where $\omega_{impl} \approx 0.15$ represents the efficiency reduction rate due to implementation overhead (about 15%).
Security Verification Cost
In systems spanning multiple layers, additional verification steps are needed for security:
$E_{security} = E_{actual} \cdot (1 – \omega_{sec})$
Where $\omega_{sec} \approx 0.1$ represents the additional efficiency reduction rate due to security verification (about 10%).
Network Latency and Coordination Cost
Additional costs arise from inter-layer communication and state coordination:
$E_{final} = E_{security} \cdot (1 – \omega_{coord})$
Where $\omega_{coord} \approx 0.2$ represents the additional efficiency reduction rate due to network latency and coordination (about 20%).
Realistic Efficiency Estimation
Considering all these realistic constraints:
$E_{final} \approx 2,281 \times (1 – 0.15) \times (1 – 0.1) \times (1 – 0.2) \approx 1,393$
Therefore, the efficiency improvement that can be expected in actual implementation is about 1,393 times, with a gas cost reduction rate of approximately 99.93%.
Complex State Synchronization Scenario Analysis
In real RWA state synchronization scenarios, the complexity of state synchronization can increase significantly. To model this situation, we introduce the state complexity factor $c$:
$S_{complex}(n, d, c) = \xi \cdot n \cdot (1 + \log(d)) \cdot c$
In typical financial RWA scenarios, $c \approx 2.5$ is estimated. The final efficiency in such complex state synchronization scenarios is:
$E_{final,complex} \approx \frac{E_{final}}{c} \approx \frac{1,393}{2.5} \approx 557$
This corresponds to an efficiency improvement of about 557 times compared to L1 in complex RWA state synchronization scenarios, i.e., a gas cost reduction of approximately 99.82%.
Relationship Between Batch Size and Efficiency
Efficiency can vary greatly depending on batch size. The following graph shows the change in efficiency with L3 batch size ($n$) and L2 batch size ($m$):
As shown in the graph, efficiency improves as batch size increases, but diminishing returns occur beyond a certain size. This is because as batch size increases, the rate of efficiency gain decreases due to the nature of batch processing.
Conclusion and Implications
In this study, we analyzed the gas efficiency provided by Oraclizer’s L3 zkRollup architecture through mathematical modeling. The theoretical analysis shows that gas cost reductions of about 99.8% are possible even in complex state synchronization scenarios.
This high efficiency stems from the combined effects of L3 layering, the Validium approach, and the zkVerify network, with each technological element contributing to efficiency improvement through different mechanisms. Specifically, L3 layering provides about 99.6% gas reduction compared to L1, Validium provides an additional 78.7% reduction, and zkVerify provides an additional 64.3% reduction. These individual effects work together to determine overall efficiency.
In actual implementation, various constraints may lead to lower efficiency than the theoretical maximum, but nonetheless, very significant efficiency improvements over existing solutions can be achieved.
The results of this study suggest that the Oraclizer architecture can make complex tasks like state synchronization economically feasible. This enables the scaling of innovative applications such as real-world asset tokenization, ultimately promoting the practical adoption of blockchain technology.
Future research should validate this theoretical model by comparing it with testnet and actual implementation data, and more accurately measure performance in various state synchronization scenarios.
References
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- Matter Labs. (2022). zkSync 2.0: A ZK Rollup For Ethereum With EVM Compatibility.
- Bünz, B., Kiffer, L., Luu, L., & Zamani, M. (2019). FlyClient: Super-Light Clients for Cryptocurrencies. IEEE Symposium on Security and Privacy (SP).
- Thibault, S., Baccam, J., Somé, M., & Benoit, V. (2023). A comparative study on zk-rollups. In Proceedings of the 3rd Conference on Blockchain Research & Applications for Innovative Networks and Services (BRAINS).
