ZKProphet: Understanding Performance of Zero-Knowledge Proofs on GPUs

Table of Contents

1. Introduction

Zero-Knowledge Proofs (ZKPs) represent a revolutionary cryptographic protocol that enables one party (the Prover) to demonstrate knowledge of a secret input without revealing any information about the secret itself. This capability has enabled transformative applications in private cryptocurrencies, verifiable computation outsourcing, and blockchain scaling solutions. The fundamental challenge in ZKP adoption has been the substantial computational overhead required for proof generation, which can take several minutes on modern CPUs for complex computations.

GPUs have emerged as the primary acceleration platform for ZKPs due to the data-parallel nature of core computational kernels. As shown in Figure 1, GPU-accelerated ZKPs demonstrate up to 200x speedup compared to CPU implementations. However, despite these impressive gains, a systematic characterization of performance bottlenecks and scalability limitations on modern GPU architectures has been notably absent from the literature.

2. Background and Related Work

2.1 Zero-Knowledge Proof Fundamentals

Zero-Knowledge Proofs operate on the principle that a Prover can convince a Verifier of knowledge of a witness $w$ for a public function $f$ and input $x$ such that $f(x,w) = y$, without revealing any information about $w$. The Groth16 protocol, which forms the basis of this study, provides succinct proofs and sub-millisecond verification times, making it particularly suitable for real-world applications.

2.2 GPU Acceleration in Cryptography

Previous work in GPU acceleration of cryptographic primitives has demonstrated significant performance improvements. Studies like [19,30,31,42] have shown that the parallel architecture of GPUs is well-suited for cryptographic operations, particularly those involving large-scale mathematical computations. However, these efforts have primarily focused on individual kernels rather than end-to-end system performance.

3. Methodology and Experimental Setup

3.1 ZKProphet Framework

ZKProphet provides a comprehensive analysis framework for evaluating ZKP performance on GPUs. The framework systematically evaluates core computational kernels including Multi-Scalar Multiplication (MSM) and Number-Theoretic Transform (NTT), which collectively account for over 95% of the computational workload in ZKP generation.

3.2 Benchmark Configurations

Our experimental setup utilizes modern GPU architectures from NVIDIA's Ampere and Ada Lovelace generations. We evaluate performance across varying constraint counts, which represent the complexity of the computation being proved. The benchmarks include both synthetic workloads and real-world ZKP applications from cryptocurrency and blockchain domains.

4. Performance Analysis Results

4.1 Kernel Performance Breakdown

Our analysis reveals a critical shift in performance bottlenecks. While prior research focused heavily on optimizing MSM operations, we find that with modern optimized MSM implementations, NTT operations now account for up to 90% of proof generation latency. This represents a significant paradigm shift that requires renewed focus on NTT optimization.

Figure 1: Prover Speedup vs Number of Constraints
The experimental results demonstrate that GPU implementations provide substantial speedup over CPU baselines, with performance scaling approximately linearly with constraint count up to 200x improvement for large constraint sets.

4.2 Scalability Analysis

We observe that ZKP computations execute exclusively on GPU's 32-bit integer pipelines and exhibit limited instruction-level parallelism due to inherent data dependencies. This fundamentally limits performance scaling based on available integer compute units rather than floating-point capabilities.

5. Technical Implementation Details

5.1 Mathematical Foundations

The core mathematical operations in ZKPs rely on finite field arithmetic over large prime fields. The NTT operation, which is a specialized form of Fourier transform in finite fields, can be expressed as:

$$X_k = \\sum_{j=0}^{n-1} x_j \\cdot \\omega^{jk} \\mod p$$

where $\\omega$ is a primitive $n$-th root of unity modulo $p$, and $p$ is a large prime. The inverse NTT is similarly defined with $\\omega^{-1}$.

5.2 Code Implementation

The following pseudocode demonstrates an optimized NTT implementation for GPU architectures:

__global__ void ntt_kernel(uint32_t *a, uint32_t *roots, int n) {
    int tid = threadIdx.x + blockIdx.x * blockDim.x;
    int stride = blockDim.x * gridDim.x;
    
    for (int i = tid; i < n/2; i += stride) {
        int j = bit_reverse(i, log2(n));
        if (i < j) {
            swap(a[i], a[j]);
        }
    }
    
    __syncthreads();
    
    for (int len = 2; len <= n; len <<= 1) {
        int half = len >> 1;
        for (int i = tid; i < n; i += stride) {
            if ((i & (len - 1)) < half) {
                uint32_t u = a[i];
                uint32_t v = mul_mod(a[i + half], roots[len + (i & (half - 1))]);
                a[i] = add_mod(u, v);
                a[i + half] = sub_mod(u, v);
            }
        }
        __syncthreads();
    }
}

6. Future Applications and Directions

The optimization of ZKPs on GPUs opens numerous application possibilities. In blockchain technology, optimized ZKPs can enable more efficient rollup solutions and privacy-preserving smart contracts. For verifiable machine learning, as explored in works like [32], GPU-accelerated ZKPs could enable practical verification of model inferences without revealing proprietary model parameters.

Future research directions include exploring mixed-precision arithmetic to better utilize GPU tensor cores, developing specialized hardware-software co-design approaches, and creating adaptive optimization frameworks that can automatically tune ZKP parameters based on specific hardware capabilities and application requirements.

Original Analysis

The ZKProphet study represents a significant advancement in understanding the performance characteristics of Zero-Knowledge Proofs on modern GPU architectures. While previous research, such as the foundational work on zk-SNARKs by Ben-Sasson et al. (2014), established the theoretical foundations, and subsequent implementations like libsnark and bellman provided practical frameworks, there has been a notable gap in systematic performance analysis across the entire computational pipeline.

The identification of NTT as the new primary bottleneck (accounting for up to 90% of latency) marks a critical shift in optimization priorities. This finding aligns with observations in other compute-intensive domains where initial optimizations target the most obvious bottlenecks, only to reveal secondary constraints that become dominant after initial improvements. Similar patterns have been observed in cryptographic implementations for blockchain systems, where after optimizing elliptic curve operations, memory access patterns became the limiting factor.

The exclusive use of 32-bit integer pipelines presents both challenges and opportunities. Unlike machine learning workloads that heavily utilize tensor cores and FP32/FP16 arithmetic, ZKPs cannot benefit from these specialized units in current GPU architectures. This suggests potential for hardware-software co-design, similar to the approach taken in Google's TPU architecture for neural networks, but specialized for cryptographic operations. The limited instruction-level parallelism due to data dependencies further emphasizes the need for algorithmic innovations that can expose more parallelism.

Compared to other cryptographic acceleration efforts, such as those for homomorphic encryption (as referenced in [9]) or verifiable computation frameworks, ZKProphet's focus on end-to-end performance rather than individual kernel optimization provides more practical insights for real-world deployment. The reference to verifiable machine learning applications in [32] suggests promising cross-domain applications where GPU-accelerated ZKPs could enable new trust models in AI systems.

The performance scalability limitations identified in this work have significant implications for the practical deployment of ZKPs in production systems. As constraint counts increase with more complex computations, the linear scaling relationship suggests that current GPU architectures may face fundamental limits without architectural innovations specifically targeting cryptographic workloads.

7. References

1. Groth, J. (2016). "On the Size of Pairing-Based Non-interactive Arguments." EUROCRYPT 2016.
2. Ben-Sasson, E., et al. (2014). "Succinct Non-Interactive Zero Knowledge for a von Neumann Architecture." USENIX Security Symposium.
3. Parno, B., et al. (2013). "Pinocchio: Nearly Practical Verifiable Computation." IEEE Symposium on Security and Privacy.
4. Setty, S., et al. (2013). "Resolving the conflict between generality and plausibility in verified computation." EuroSys.
5. Zhang, J., et al. (2020). "vCNN: Verifiable Convolutional Neural Network based on zk-SNARKs." Cryptology ePrint Archive.
6. Wahby, R.S., et al. (2018). "Full accounting for verifiable outsourcing." CCS.
7. Kosba, A., et al. (2016). "C∅C∅: A Framework for Building Composable Zero-Knowledge Proofs." USENIX Security.
8. Xie, T., et al. (2022). "zkCNN: Zero Knowledge Proofs for Convolutional Neural Network Predictions and Accuracy." CCS.