Parallel Mira-Based Proof Accumulation

• The paper introduces a tree-based, parallelizable extension of Mira accumulation that reduces sequential bottlenecks and compresses proof sizes effectively.
• The methodology restructures cryptographic proof folding into logarithmic depth, achieving up to 6.2× speedups and significant scalability improvements.
• This technique is pivotal for zero-knowledge ML and distributed computation, enabling efficient batch verification and robust proof aggregation.

Parallel Mira-Based Proof Accumulation is an advanced technique for constructing, verifying, and compressing sequences of cryptographic proofs using a fully parallelizable extension of the Mira accumulation protocol. The core idea is to reduce the inherent sequential bottleneck of previous proof accumulation schemes and achieve significant gains in proof size, scalability, and wall-clock proving time by restructuring the accumulation as a tree-based, homomorphic fold compatible with parallel computation. This approach has direct applications in zero-knowledge machine learning, distributed computation verification, and proof systems that require efficient and robust batch composition. Recent frameworks, including ZKTorch, demonstrate up to 6× end-to-end speedups and 3×–10× reductions in proof size compared to prior methods (Chen et al., 9 Jul 2025).

1. Foundational Principles of Mira Accumulation

The Mira scheme is designed for succinct non-interactive arguments of knowledge (NARKs) in pairing-based proof systems. Each basic proof instance consists of public input commitments $[C_i]_{i=1}^k$, witness openings $[m_i]_{i=1}^k$, Fiat-Shamir (FS) challenges $[r_i]_{i=1}^{k-1}$, slack variables $\mu$, and an algebraic error commitment $E$ associated to an error value $e$ in the scalar field.

The essential operation is the random linear combination (RLC) fold: $$(v_0,E_0,e_0),\;(v_1,E_1,e_1) \longmapsto (\gamma v_0 + v_1,\;E_1 + \gamma E_0,\;e_1 + \gamma e_0),$$ where $\gamma$ is a randomly sampled scalar from a random oracle applied to the two inputs. This process produces a new single accumulator instance embodying the combined proof-checking constraints.

Traditionally, accumulation is performed in a sequential loop, incrementally folding each new proof into the running accumulator, resulting in $O(N)$ complexity in both size and time for $N$ proofs (Chen et al., 9 Jul 2025).

2. Parallelization: The Tree-Reduce Architecture

Parallel Mira accumulation restructures the sequence of folds into a binary tree, enabling $O(\log N)$ depth and allowing simultaneous folding of multiple proof pairs in each round. This is achieved by treating every leaf proof as an independent accumulator and recursively combining halves of the accumulator list in parallel.

Pseudocode for the parallel fold:

def ParallelAccumulate(list_L): if len(list_L) == 1: return list_L[0] # Split list in half L0, L1 = split(list_L) # Parallel recursive accumulation A0 = ParallelAccumulate(L0) A1 = ParallelAccumulate(L1) return Fold(A0, A1)

Each Fold(A0, A1) operation is a pure homomorphic combination on the underlying group elements and scalars, and the associativity/commutativity of the RLC ensures that the final result is independent of tree shape, except for negligible soundness errors due to random oracle binding.

This parallelization reduces wall-clock accumulation time by a $1/N$ factor for $N$ available cores, with the total proof size shrinking dramatically to $O(\log N)$. For large $N$, the final accumulator remains effectively constant in size (≤10 group elements for $N\leq 10^6$) (Chen et al., 9 Jul 2025).

3. Algorithmic Properties and Correctness

Every binary fold in parallel Mira preserves the relaxed algebraic test

$\mathcal{V}_{\mathrm{lin}}(v,e) = 0,$

maintaining the soundness property of the underlying NARK protocol. Knowledge-soundness is maintained via the special-soundness of the $(d+1)$-fold Fiat-Shamir protocol underpinning each block.

A Vandermonde-based extractor ensures that any grouping of inputs yields the identical final accumulator up to negligible binding errors, and as a consequence, any adversarial manipulation of intermediate states cannot escape the soundness constraints of the protocol.

4. Integration into ZKTorch and ML Inference

In practice, ZKTorch decomposes machine learning models into basic blocks (Add, Mul, MatMul, etc.), each associated with disjoint NARK proofs. Each such proof is dispatched as a leaf in the parallel Mira tree-reduction. Only blocks compatible with the relaxed algebraic-test form are accumulated in parallel; others are verified via Poly-IOP and merged at the root.

Leaves are processed in parallel across available cores, and the accumulation is performed in $\lceil\log_2 N\rceil$ synchronization rounds. Constraints on degree $d$ affect error vector size and per-fold communication; memory requirements scale with the number of leaves and tree depth.

Empirical benchmarks from ZKTorch show that for large models (GPT-j, BERT, ResNet-50, LLaMA-2-7B), parallel Mira yields up to 6.2× speedup in proving time and proof sizes reduced 3×–10× compared to prior specialized schemes (Chen et al., 9 Jul 2025). Representative timings:

Model	$t_{\mathcal{P}}$ Seq	$t_{\mathcal{P}}$ Par (speedup)	Proof Size (MB)
GPT-j	8663 s	1398 s (6.2×)	~6
BERT	1118 s	880 s (1.3×)	~5
ResNet-50	12313 s	6271 s (2.0×)	~6
LLaMA-2-7B	5976 s	2646 s (2.3×)	~5
RNNT	36 s	19 s (1.9×)	<0.1

5. Generalization and Comparative Context

The parallel Mira-based approach generalizes prior proof-accumulation frameworks in two respects:

1. It removes the sequential dependency inherent in accumulation, permitting orders-of-magnitude speedup for large batches.
2. It yields strong compression in proof size, making cross-block batch-proof composition feasible for high-throughput ML and computation audits.

A plausible implication is that systems currently bottlenecked by sequential proof composition—such as batch verification in NARK-based zero-knowledge ML or distributed computation integrity protocols—can incorporate parallel Mira-based accumulation for substantial practical efficiency gains.

6. Limitations, Trade-offs, and Extensions

Parallel Mira is inherently compatible only with blocks that admit the relaxed algebraic-test form. Blocks requiring alternative verification (such as Poly-IOP) must be processed separately and merged post hoc. While tree depth and memory scale logarithmically for most practical scenarios, the largest deployments may require careful management of accumulator states.

Mira-based accumulation is fundamentally limited by the trusted setup and cryptographic assumptions of the underlying pairing-based SNARKs and random oracle models. Trade-offs in degree selection can affect accumulators' size and communication overhead, which must be optimized per application.

Extensions include the combination of parallel Mira-based techniques with distributed proof generation frameworks, such as Camelot’s Reed–Solomon-encoded batch evaluation protocols (Björklund et al., 2016), providing Byzantine-robust error correction, distributed decoding, and randomized local verification.

7. Significance in Cryptographic and Distributed Settings

Parallel Mira-based proof accumulation represents a convergence of cryptographic batching methods and parallel verification protocols, exemplified in frameworks such as ZKTorch (Chen et al., 9 Jul 2025) and distributed computation platforms like Camelot (Björklund et al., 2016). This approach offers a scalable pathway for succinct, robust, and independently verifiable proofs in high-throughput systems, addressing bottlenecks in both zero-knowledge ML and distributed algorithm auditability. These techniques support optimal total work, near-perfect parallel speedup, and Byzantine fault tolerance, constituting a robust foundation for modern proof systems in both cryptography and distributed computation.