Halo2 Proving Systems

• Halo2 proving systems are zero-knowledge frameworks that encode batched state transitions into arithmetic circuits for ZK-rollups.
• They employ a multi-stage architecture—including super, batch, and bundle circuits—to recursively aggregate proofs and optimize throughput.
• Integrated parametric cost models and Z3 SMT solvers enable efficient resource allocation, reducing operational costs while ensuring prompt finality.

Halo2-based proving systems are a class of zero-knowledge (ZK) succinct proof frameworks developed for proving statement validity in large-scale cryptographic protocols, notably in the context of zero-knowledge rollups (ZK-rollups). In these systems, complex statements—such as batched state transitions in a rollup—are encoded as large arithmetic circuits, processed through a hierarchical pipeline of specialized prover components. Economic viability, system finality, and off-chain/on-chain cost minimization are primary concerns when deploying Halo2-based systems at high throughput and under dynamic load, prompting research on rigorous modeling and optimization methodologies.

1. Architecture of Halo2 Proving Systems

Halo2 encodes computational statements as arithmetic circuits constructed from custom gates and organized in a table of polynomial constraints, where each constraint enforces relations on “wires” of the circuit. These wires are divided into advice, fixed, and lookup columns. The circuit’s total provable capacity—i.e., the maximum computational workload for which a single proof can be generated—is determined by the constraint table’s row capacity (bounded by the polynomial commitment scheme, e.g., $2^{28}$ rows for KZG).

Proof generation in the setting of rollups proceeds in multiple stages:

• Super circuits: Process chunks of incoming transactions, each mapping to one circuit invocation.
• Batch circuits: Aggregate multiple super-circuit proofs.
• Bundle circuits: Recursively aggregate batch proofs to produce a final succinct proof.

The multi-stage design is critical for scaling: by batching and recursively aggregating transactions off-chain, Halo2-based systems maintain both efficiency and security (zero-knowledge and succinct verification) for on-chain verification.

Parameter selection—including the number of constraint rows, batch sizes, recursion depth, and epoch timing—directly influences system throughput and finality.

2. Parametric Cost Model and Constraint System

The cost and performance of a Halo2-based proving infrastructure in a ZK-rollup are captured via a parametric model, which explicitly models the dependencies between:

• Transactions Per Second (TPS)
• Average Gas Usage per Transaction ($Gas_{PerTx}$)
• Target Finality ($T_{\text{finality}}$)

Hardware and protocol constants, such as per-machine cost, circuit times at each proving stage, and maximum circuit capacities, further parameterize the model.

For example, the required number of super provers is calculated as: $$N_{\text{super}} = \left\lceil \frac{\text{TPS} \cdot T_{\text{super}}}{\text{Tx}_{\text{max}}} \right\rceil,$$ where $T_{\text{super}}$ is the time per super proof, and $\text{Tx}_{\text{max}}$ is the per-prover maximum transaction load.

The batch epoch, governing batching interval, is defined by: $$\text{Epoch}_{\text{batch}} = \min \left( \max \left( T_{\text{super}}, T_{\text{batch}}, \left\lceil \frac{\text{Max}_{\text{super proofs per batch}} \cdot T_{\text{super}}}{N_{\text{super}}} \right\rceil \right), T_{\text{finality}} - T_{\text{super}} - T_{\text{bundle}} \right).$$

Number of batch provers and their work allocation similarly follow: $$N_{\text{batch}} = \left\lceil \frac{N_{\text{super proofs per epoch}}}{\text{Max}_{\text{super proofs per batch}} \cdot \left\lfloor \frac{\text{Epoch}_{\text{batch}}}{T_{\text{batch}}} \right\rfloor} \right\rceil.$$

Cost modeling includes computation for on-chain Data Availability (DA) and verification. Using blob-based DA as an example, batch costs are: $$\text{Cost}_{\text{DA per batch}} = \text{Gas}_{\text{DA per batch}} \times \text{Price}_{\text{gas (Gwei)}} \times 10^{-9} \times \text{Price}_{\text{ETH/USD}}.$$ Monthly cost sums hardware, DA, and verification costs over all batches and bundles.

3. Z3 SMT Solver for Optimal Resource Allocation

The non-linear interdependencies—such as those between TPS, hardware provisioning, and batching epochs—necessitate a constraint programming approach. These dependencies are encoded using the Z3 SMT solver, which is operated programmatically via its Python API:

• An optimization context is established (e.g., opt = Optimize()).
• Parallelization parameters are tuned (e.g., z3.set_param('parallel.threads.max', 64)).
• Integer decision variables (provers per stage) and real-valued epoch durations are instantiated.

Constraints enforce:

• At each layer, the provisioned throughput must meet or exceed the incoming transaction load;
• Total system latency across super, batch, and bundle proofs must remain under $T_{\text{finality}}$;
• Correct ordering between pipeline stages.

The overall cost objective minimized is: $$\text{total\_cost} = \sum_{\ell \in \{\text{super, batch, bundle}\}} N_\ell \cdot C_\ell + (\text{monthly\_batches} \times \text{DA\_USD}) + (\text{monthly\_bundles} \times \text{Verif\_USD}),$$ where $N_\ell$ and $C_\ell$ are the number of provers and cost per prover for each pipeline stage.

Z3 automatically searches for minimal-cost parameter configurations that ensure both compliance with performance constraints (including lag avoidance and finality) and optimal economic provisioning.

4. Discrete-Event Simulator for Empirical Validation

A detailed discrete-event simulator is implemented to verify the robustness of Z3-derived configurations under realistic conditions. The simulation models two interleaved epochs (batch and bundle) and tracks resource flow through the proof pipeline:

• Outputs of super provers build a “proof pool” consumed by batch provers and, subsequently, bundle aggregators.
• The simulator updates system state at each epoch boundary.
• Lag detection is performed, flagging unscheduled resource accumulation (e.g., overflow of super proofs unprocessed by batch provers).

In workload replays (e.g., 30 days of transaction data), the optimal configurations produced by the Z3 solver are shown to sustain the required finality (such as within a 4-hour deadline) and exhibit no pipeline lag. This confirms the constraint model’s fidelity in capturing real bottlenecks and resource utilization.

5. Economic and Operational Findings

Empirical evaluation reveals substantial cost improvement over naïve dynamic sizing strategies. Specifically, up to 70% total operational cost reduction is possible under steady load: monthly cost decreased from $74K to$22K under the optimized fleet. These savings are principally achieved through fine-grained tuning of batch and bundle epochs and accurate sizing of prover fleets at all stages, thereby preventing over-provisioning.

The model’s adaptability to dynamic load and volatile gas prices suggests that automated reasoning approaches (e.g., Z3-based scheduling) can support near-real-time infrastructure adjustments, directly contributing to the economic efficiency and scalability of Layer 2 protocols.

6. Implications for Scalability and Rollup Viability

The combination of parameteric cost modeling, SMT-based optimization, and discrete-event simulation substantiates the case that Halo2-based proving systems can be deployed with predictable cost, robust to performance demands and operational volatility. Lowering the amortized cost of proof generation enhances the rollup’s competitiveness relative to Layer 1 fees; maintaining strict constraints on finality and data availability compliance is vital for ecosystem trust and adoption.

Broader adoption of these modeling and optimization strategies may further stimulate advances in ZK-rollup architectures, enabling higher throughput and supporting increasingly complex decentralized applications without compromising economic viability or security.