Zero-Knowledge Machine Learning (ZKML)

• Zero-Knowledge Machine Learning (ZKML) is a framework that combines zero-knowledge proofs with machine learning to validate model outputs without revealing internal data.
• It employs collision-resistant hashing, polynomial fingerprinting, and zk-STARK protocols to ensure cryptographic integrity and auditability in high-throughput settings.
• ZKML techniques enable real-time, privacy-preserving inference in critical applications such as particle physics, regulated finance, and on-chain AI through efficient hardware acceleration.

Zero-Knowledge Machine Learning (ZKML) is the synthesis of advanced cryptographic proof techniques—primarily zero-knowledge proofs (ZKPs)—with machine learning workflows, enabling the verification of key ML operations without revealing sensitive data, model internals, or proprietary computation traces. In high-assurance applications such as particle physics, regulated financial systems, and on-chain AI services, ZKML ensures cryptographic integrity, auditability, and privacy for inference, training, and fairness verification. Recent frameworks such as PHAZE at the LHC instantiate ZKML to meet extreme throughput and latency demands, certifying decisions via succinct proofs and domain-specific hashing protocols (Jawahar et al., 16 Nov 2025). This article systematically presents the essential architecture, proof mechanisms, cryptographic primitives, security properties, efficiency optimizations, and illustrative deployments of ZKML, particularly as realized in PHAZE.

1. Core Principles and ZKML Workflow

The defining objective of ZKML is to cryptographically assure that a model output—be it a prediction, trigger classification, or fairness metric—was derived precisely as claimed, using proprietary or sensitive assets, while leaking no auxiliary information. The standard ZKML pipeline decomposes into:

• Model Commitment: The full set of model parameters $\theta$ is bound by a succinct, collision-resistant hash $C \leftarrow H(\theta)$, serving as the model commitment.
• Offline Proof Construction: For each training datum $x_i$, full inference $y_i = f_\theta(x_i)$ and intermediate (early-exit) activations $A_i = EE_m(x_i)$ are computed. These activations are quantized, interpolated (to obtain $P_i \in \mathcal{P}_d(\mathbb{F}_p^m)$), and fingerprinted ($h_i = P_i(r)$, Rabin fingerprint at public challenge $r$).
• Zero-Knowledge Proof Generation: Computational integrity for $(C, x_i, y_i, P_i, h_i)$ is arithmetized in AIR and proven by zk-STARK, establishing $$C = H(\theta) \wedge y_i = f_\theta(x_i) \wedge P_i$$ interpolates $A_i^* \wedge P_i(r) = h_i$.
• Online Phase: For each new event $x_{\text{new}}$, early-exit activations $A_{\text{new}}$ are extracted (FPGA, $O(100-200\,\mathrm{ns})$), quantized and interpolated, and the fingerprint $h_{\text{new}}$ computed ($\sim$50 ns). $h_{\text{new}}$ is instantly mapped via VDM ($O(2.5\,\mathrm{ns})$) to the certified decision $D'$, or triggers anomaly detection if unrecognized.

PHAZE's architecture achieves high-throughput, sub-microsecond inference verification by precomputing the maximal proof burden offline and condensing online logic into bounded cryptographic hash and table operations (Jawahar et al., 16 Nov 2025).

2. Cryptographic Hashing and Polynomial Fingerprinting

Central to ZKML's fast verification is the use of collision-resistant hash functions and polynomial fingerprinting:

• Rabin Fingerprinting: Quantized activation vector $A^* \in \mathbb{F}_p^k$ is encoded as a degree-$d$ polynomial $P(x)$ over $\mathbb{F}_p$. Fingerprint $h = P(r)\bmod g(x)$ (irreducible $g(x)$, random $r$) compactly binds high-dimensional activations to a 64-bit key, exploitable for constant-time lookups.
• Collision Bounds: By the Schwartz–Zippel lemma, $\Pr[P_1(r) = P_2(r)] \leq (d-1)/|\mathbb{F}_p|$; empirically $\varepsilon \approx d / 2^{64}$. Estrin’s method allows efficient O($d$) computation on modern FPGAs at $\sim$50 ns per event.

The fingerprinting module provides both fast, probabilistic binding and a latent anomaly detection mechanism whenever $h_{\text{new}}$ does not resolve in the VDM.

3. Formal Zero-Knowledge Proof Construction

The computational integrity and privacy of inference outcomes are certified by algebraic zero-knowledge systems:

• NP Relation:

$$R = \left\{ (C, x, y, P, h;\theta) \;|\; C = H(\theta) \wedge y = f_\theta(x) \wedge P \text{ interpolates early activations} \wedge P(r) = h \right\}$$

• zk-STARK Protocol:
  • Key generation requires only public randomness (no trusted setup).
  • Prover arithmetizes computation in AIR constraints, commits polynomials in Merkle trees, and outputs proof $\pi$ in $O(T \cdot \mathrm{poly\,log} T)$, $T$ model size.
  • Verification is $O(\mathrm{poly\,log} T)$ (typically ms-scale).
• Proof Size: $O(\mathrm{poly\,log}\, T)\cdot \kappa$ bits, a few hundred KB for million-parameter models.

The proofs link the model commitment to certified activations, enforcing integrity without revealing model internals or intermediate activations.

4. Low-Latency, Certifiable Early-Exit Mechanisms

To comply with stringent real-time constraints—as at the LHC ($40\,\text{MHz}$, $200\,\text{ns}$ budget)—PHAZE implements a certifiable early-exit:

• Early-Exit Criteria:
  • Early-layer activations $A$ must admit a trigger decision $y \approx f_{\text{full}}(x)$ with negligible loss of fidelity.
  • The proof system binds these activations to the full offline model and its commitment.
• Latency Breakdown:

$$T_{\text{online}} = T_\text{EE} + T_\text{hash} + T_\text{lookup} \approx 152.5\,\text{ns} \text{ to } 252.5\,\text{ns}/\text{event}$$

| Stage | Time (ns) | Hardware | Complexity | |----------------------|-------------|----------|--------------------| | EE activation | 100–200 | FPGA | $O(|\mathbb{M}_\text{early}|)$ | | On-the-Fly hashing | 50 | FPGA | $O(\sqrt{d})$ | | VDM lookup | 2.5 | FPGA | $O(1)$ |

This enables massive parallelization (multiple FPGAs, O(40 MHz)), meeting extreme throughput requirements for event selection.

5. Security Properties: Integrity, Anomaly Detection, and Privacy

PHAZE provides strong security guarantees:

• Anomaly Detection: A lookup miss $(h_\text{new} \notin \text{VDM})$ flags either unmodeled physics or hardware malfunction, with no additional computational overhead.
• Threat Model: Tampering at the hardware/memory layers cannot forge a decision mapping without breaking the hash function $H$ or the zk-STARK soundness.
• Zero-Knowledge: No private parameter $\theta$ or early activations $A^*$ are leaked in the course of proof, providing data/model privacy, albeit secondary to integrity in PHAZE.

Completeness and soundness are formally established: correct events are always accepted, and forgeries succeed only with negligible $\varepsilon$.

6. Evaluation and Domain-Specific Deployment

Empirical benchmarking confirms the viability of ZKML in physics-grade environments:

• Build-Phase (zk-proof, MNIST, 7M params):
  • Proof generation (EZKL): $~100\,\text{s}$ per event (unoptimized)
  • Verification: $1-10\,\text{ms}$
  • Fingerprinting: $1-10\,\mu\text{s}$
• FPGA Online Throughput:
  • $~4-6\ \text{MHz}$ per FPGA, parallelizable to $40\,\text{MHz}$
  • One AMD UltraScale+ ($455\,\text{Mb}$ BRAM) can store $6.3 \times 10^6$ 64-bit hash entries

Meeting the LHC trigger constraint ($40\,\text{MHz}$) is achieved by instantiating parallel lanes and distributed lookup infrastructures. The framework allows dynamic reconfiguration and scales beyond individual hardware limitations.

7. Broader Implications and Future Directions

ZKML, as exemplified by PHAZE, demonstrates the feasibility of cryptographically sound, privacy-preserving inference at unprecedented latency and throughput (Jawahar et al., 16 Nov 2025). The methodology extends to domains requiring real-time integrity with rapid anomaly detection and is compatible with post-quantum, transparent proof systems.

Potential extensions include:

• Distributed dynamic triggering in future collider experiments
• Built-in protocol for non-equilibrium anomaly detection in physics streams
• Cross-domain adaptation to privacy-preserving compliance audits and financial risk models (Scaramuzza et al., 30 Oct 2025)

By coupling cryptographic model commitment, polynomial fingerprinting, and early-exit inference, ZKML advances both the theoretical basis and practical deployment of verifiable ML in high-throughput, mission-critical settings.