Polygon zkEVM Advances

• Polygon zkEVM is an advanced cryptographic system that uses zk-SNARKs to compress and validate Ethereum state transitions.
• It integrates heterogeneous circuit designs and recursive aggregation to enable scalable and privacy-preserving execution of EVM smart contracts.
• Optimization strategies, including efficient hash functions and hardware acceleration, reduce on-chain gas costs while enhancing performance.

Polygon zkEVM is an advanced cryptographic system that enables scalable, privacy-preserving execution of Ethereum Virtual Machine (EVM) smart contracts through the deployment of zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs). The system’s design compresses heterogeneous EVM state transitions—encompassing transaction validation, storage, and computation—into succinct proofs which can be efficiently verified on-chain. This underpins scalable validity rollups for Ethereum-compatible blockchains, facilitating higher throughput without compromising decentralization or privacy.

1. Cryptographic Foundations: zk-SNARKs and Arithmetic Circuits

At the heart of zkEVM lies the representation of EVM state transitions as arithmetic circuits. Any EVM computation is first encoded into an arithmetic circuit: wire assignments must satisfy both multiplication and addition constraints. This encoding is then transformed into a Quadratic Arithmetic Program (QAP), which reduces the correctness of a computation to a polynomial divisibility condition:

$$\sum_{i=1}^n s_i u_i(x) \cdot \sum_{i=1}^n s_i v_i(x) - \sum_{i=1}^n s_i w_i(x) \equiv 0 \pmod{t(x)}$$

where $s$ is the witness (circuit wire assignments), $u_i(x), v_i(x), w_i(x)$ are polynomials marking different gates, and $t(x)$ is the target (vanishing) polynomial defined over the circuit’s constraints (Chen et al., 2022). This divisibility check ensures that all circuit constraints are satisfied.

Cryptographic protocols such as Pinocchio and Groth-16 map these QAP checks onto succinct group element relations using bilinear pairings. The verifier confirms, via a small set of pairing operations,

$$e(E(u(z)), E(v(z))) = e(E(w(z)), E(1)) \cdot e(E(t(z)), E(h(z)))$$

where $E(x)$ encodes group elements and $e$ is the pairing function.

2. Privacy and Scalability Through zk-Rollups

zkEVM leverages zk-SNARKs to encode a complete block’s state transition as a single proof, hiding all transaction data and internal computation (“the witness”) off-chain. Only succinct group elements and a state root are visible, preserving privacy. On-chain verification of these proofs is constant in complexity (and independent of the block size) (Chen et al., 2022).

For scalability, this “rollup” of thousands of transactions into a single constant-sized proof shifts computational burden off-chain and minimizes data on Ethereum's Layer 1. Validity rollups guarantee correctness of state updates while supporting aggressive batching, optimizing on-chain gas expenditures.

3. Circuit Design and Recursive Aggregation Strategies

Implementing zkEVM presents considerable circuit engineering challenges due to EVM opcode diversity and cost heterogeneity. There are two principal circuit architectures:

• Naive TinyRAM Approach: Compilation of EVM bytecode to generic TinyRAM circuits is straightforward but incurs severe overhead (up to $1000\times$ more gates for operations like array access).
• Heterogeneous Mixing: zkEVM implementers combine hardwired specialized circuits (for expensive opcodes such as SSTORE, signature verification, hashing) with TinyRAM for low-cost opcodes. These heterogeneous circuits are blended using recursive SNARK aggregation, staging proofs for different components and aggregating them recursively to maintain a constant final proof size (Chen et al., 2022).

Protocols such as PLONK support efficient recursive aggregation over these mixed circuits. The practical impact is a scalable system that maintains succinct proof sizes and supports many types of EVM transactions simultaneously.

4. Trusted Setup, Transparency, and Protocol Innovations

Traditional zk-SNARKs (e.g., Groth-16) rely on a trusted setup (the “Power of Tau” ceremony) to generate common reference strings, introducing potential security risks. The computational cost of trusted setup grows exponentially with the security parameter $n$:

$\text{CRS} = \{\tau, \tau^2, \ldots, \tau^n\}$

For $n=15$, setup may take minutes; for $n=19$ or $n=20$, time and memory consumption can be prohibitive (Habib, 21 Mar 2025).

The LUMEN protocol introduces transparent, efficient zk-SNARKs—using recursive polynomial commitment schemes and polynomial interactive oracle proofs—with proof sizes near $1\text{KB}$, competitive runtime, and no trusted setup requirement (Quan, 2023). LUMEN’s techniques (use of groups with hidden orders, Lagrange basis polynomials, amortization strategies) deliver strong completeness, soundness, and zero-knowledge guarantees, and are implemented in approximately 8000 lines of Rust and Python. This fosters trustless deployment and removes centralized security assumptions for Polygon zkEVM integration.

5. Hash Functions, Merkle Trees, and Practical Benchmarking

Efficient hash functions are essential for Merkle tree constructions in ZK circuits, which are core components for batching, privacy mixers, and rollup protocols. Benchmarking efforts on EVM-compatible blockchains situate Poseidon and Poseidon2 as superior ZK-friendly hash functions due to lower circuit constraint counts and improved runtime efficiency when used with Groth16 SNARKs (Guo et al., 3 Sep 2024). Integrating Poseidon2 leads to a 73% reduction in on-chain gas for operations such as privacy-preserving commitments. These improvements are critical given the high “circuit power” demanded by deep Merkle trees.

Sequencer-based batching schemes further reduce on-chain costs and enable regulatory compliance; deposit commitments can be aggregated and proved off-chain, with only a single state update and proof submitted on-chain (Guo et al., 3 Sep 2024). This brings both cost-efficiency and robust compliance mechanisms to zkEVM design.

6. Performance Bottlenecks and Optimization Strategies

zkEVM scalability is limited not by proof verification—whose time remains constant—but by proof generation bottlenecks, most notably in trusted setup, circuit compilation, and SNARK key generation. For circuit batch size $B$, the constraint count $C(B)$ scales at least linearly:

$C(B) \propto B$

Proof generation time increases superlinearly with batch size, even as per-transaction gas cost decreases due to economies of scale (Habib, 21 Mar 2025).

Optimization strategies include:

• Lowering security parameter $n$ (with a corresponding trade-off in security)
• Parallelization of trusted setup and key generation
• Circuit minimization (reuse of sub-circuits, adoption of more efficient primitives)
• Deployment of specialized hardware (FPGA/ASICs) for EC pairings and cryptographic operations
• Exploration of alternative ZKP protocols, including hybrid SNARK/STARK approaches.

7. Cloud Infrastructure: ARM SVE, x86-64 AVX, and Cost–Performance Trade-offs

zkEVM’s off-chain prover performance is constrained by underlying hardware. Polygon’s zkProver is highly optimized for x86-64 platforms with AVX/AVX512 vector instructions, building large Merkle trees (over 4 million leaves) with Poseidon hashes over the Goldilocks field. The field prime is

$P = 2^{64} - 2^{32} + 1$

enabling fast 64-bit word-level arithmetic (Loghin et al., 11 Jun 2025).

ARM cloud servers equipped with SVE/SVE2 ISA offer lower cost but currently lag on performance by $1.4\times$ to $3.2\times$ (due to smaller vector width, 128 vs. 512-bit, and lower clock frequency, $\approx$3.1 GHz vs. 3.7 GHz). The time-to-prove scales as

$T = \frac{I}{IPC \cdot f}$

where $I$ is instruction count, $IPC$ is instructions per cycle, and $f$ is clock frequency. Although ARM supports unsigned arithmetic natively, advantage is negated by vector width and frequency constraints. Theoretical scaling of ARM SVE2 to 512-bit vectors and higher frequencies suggests near-parity or cost–performance advantages in future hardware designs. In the interim, optimized SVE2 assembly and parallelization can yield efficiency gains, but ultimate low-latency zkProver tasks currently favor x86-64 (Loghin et al., 11 Jun 2025).

Summary Table: Key Technical Factors and zkEVM Impact

Factor	Protocol/Implementation	zkEVM Impact
zk-SNARK (Groth-16, PLONK)	Validity rollup, recursion	Succinct proofs, batch verification
Poseidon2/Neptune hash	Merkle trees, SNARK circuits	Lower constraints, reduced on-chain cost
Trusted setup (Power of Tau)	Groth-16	Security/cost trade-off, resource bottleneck
LUMEN	Transparent SNARK	Trusted setup elimination, better security
ARM SVE/SVE2 vs. AVX512	zkProver hardware	Cost–performance trade-off, vectorization

Polygon zkEVM exemplifies the state-of-the-art in scalable, privacy-preserving Ethereum computation, sustained by continued innovation in circuit engineering, cryptographic protocol design, hardware optimization, and benchmarking of critical primitives. Future advances in transparent SNARKs (such as LUMEN), hardware vectorization (ARM SVE2), and protocol optimization will further improve the security, efficiency, and cost-effectiveness of zkEVM deployments in real-world blockchain environments.