Zero-Knowledge Model Checking

Abstract: We introduce a technology to formally verify that a software system satisfies a temporal specification of functional correctness, without revealing the system itself. Our method combines a deductive approach to model checking to obtain a formal certificate of correctness for the system, with zero-knowledge proofs to convince an external verifier that the system -- kept secret -- complies with its specification of correctness -- made public. We consider proof certificates represented as ranking functions, and introduce both an explicit-state and a symbolic scheme for model checking in zero knowledge. Our explicit-state scheme assumes systems represented as transition graphs. We use polynomial commitments to convince the verifier that the public proof certificates correspond to the secret transition relation. Our symbolic scheme assumes systems specified as linear guarded commands and uses piecewise-linear ranking functions. We apply Farkas' lemma to obtain a witness for the validity of the ranking function with public and secret components, and employ sigma protocols for matrix multiplication and range proofs to convince the verifier of the witness's existence. We built a prototype to demonstrate the practical efficacy of our two schemes on linear temporal logic verification examples. Our technology enables formal verification in domains where both the safety and the confidentiality of the system under analysis are critical.

• The paper introduces the first zero-knowledge model checking framework that verifies ω-regular properties while keeping system internals confidential.
• It combines automata-theoretic certification, cryptographic commitments, and zero-knowledge protocols using ranking functions and Farkas witnesses.
• Prototype implementations show scalable verification for both finite and infinite-state systems, highlighting its potential for regulated and safety-critical domains.

Zero-Knowledge Model Checking: Satisfying Temporal Correctness in Complete Confidentiality

Problem Statement and Motivation

The paper "Zero-Knowledge Model Checking" (2605.00487) introduces the first formal verification technology that enables a prover to convince an external verifier that an implemented system adheres to a temporal (ω-regular/LTL) specification, without revealing any operational, structural, or behavioral details of the system itself. This task is motivated by the conflicting needs in regulated or safety-critical industrial domains: regulators and customers require formal proofs of safety and functional correctness, while technology providers are incentivized to keep system designs confidential to preserve intellectual property and competitive advantage.

Classic model checking approaches necessitate disclosing the entire system’s transition structure or implementation, which is incompatible with confidentiality and privacy requirements. The authors observe that the application of zero-knowledge (ZK) proofs in verification has, surprisingly, been largely limited to propositional and first-order logic proofs; no approach has yet addressed the synthesis of zero-knowledge proofs for model checking, especially at the level of temporal logic, system structure, and certificates.

Formalization and Methodology

The proposed framework formalizes zero-knowledge model checking (ZKMC) as a protocol comprising three layered phases:

1. Formal certification: construct a proof certificate showing that the secret system ${\cal M}$ satisfies the public LTL (ω-regular) specification $\phi$, using deductive or automata-theoretic techniques.
2. Cryptographic commitment: create a commitment $c_{\cal M}$ that binds to the secret system while revealing nothing about its structure; this ensures post-hoc non-equivocation.
3. Zero-knowledge proof: employ a ZK protocol to show that the public certificate (witness) is valid with respect to the system committed in $c_{\cal M}$.

The key insight is that the proof certificate can be made public if it does not leak any structural information about the system, and that zero-knowledge techniques can be leveraged to tie this certificate soundly to a cryptographically committed, but undisclosed, system.

Model Checking Obligation

The model checking query is $\mathcal{M} \models \phi$, where $\mathcal{M}$ is a potentially infinite-state, nondeterministic discrete system, and $\phi$ is an LTL formula. Using standard automata-theoretic reductions, the specification $\phi$ is compiled into a nondeterministic Büchi automaton $\mathcal{A}_{\neg\phi}$. The model checking problem becomes equivalent to proving that the product automaton has no fair (accepting) run, i.e., that $$L_{\mathcal{M}} \cap L_{\mathcal{A}_{\neg\phi}} = \emptyset$$.

Ranking Function Certificates

The soundness proof relies on ranking functions $\phi$0, which assign a metric to every combined system and automaton state, with the following properties:

• Finite values on all initial states (initiation).
• Monotonic non-increase on all transitions (induction).
• Strict decrease on fair transitions (acceptance condition).

Such ranking functions serve as succinct, efficiently checkable certificates of correctness for ω-regular (LTL) properties.

Instantiations: Explicit-State vs. Symbolic Schemes

Explicit-State ZKMC via Polynomial Commitments

In the finite-state case, the system is encoded explicitly as a transition graph (Kripke structure), from which characteristic polynomials for the initial state set $\phi$1 and transition relation $\phi$2 are constructed. The proof obligations (initiation, induction, fairness) are contraposed: for every potential violation of the certificate, the prover must demonstrate—without revealing the actual state or transition set—that none of the violating states or transitions belong to the system.

This is achieved using KZG polynomial commitments, where the commitments and proofs are succinct (two group elements, constant in system size), and the evaluation set is public. Security (binding, soundness) relies on the polynomial commitment cryptosystem. Prover cost is quadratic in $\phi$3 (the system state space), but verification remains extremely efficient.

Symbolic ZKMC via Duality and Zero-Knowledge Linear Algebra

For large or infinite-state systems, ZKMC abstracts the state space via guarded commands over integer variables; transitions and predicates are encoded as linear constraints. Ranking functions are piecewise-affine.

The proof obligations become unsatisfiability problems for mixed (public/secret) systems of linear inequalities, which are discharged by computing integer Farkas witnesses of infeasibility—these play the role of certificates.

To preserve zero knowledge, two-tier Pedersen commitments are used for matrices and vectors, preserving row/column structure for linear algebra operations. Verification of matrix multiplications (for the duality equalities) and range proofs (showing all entries are contained in a bounded integer domain) are conducted via dedicated ZK protocols: zkMatrix [CongYY24] and Bulletproofs [bunz2018bulletproofs], integrated with bespoke range proofs and equality checks.

This symbolic scheme’s prover and verifier cost are polynomial in the number of guarded commands and automaton transitions, and independent of state space size.

Numerical and Experimental Findings

A prototype implementation validates both explicit and symbolic schemes on standard benchmarks, including models of exponential backoff handshake, DHCP, and round-robin scheduling.

• The explicit-state scheme performs efficiently up to $\phi$4 states, with proof generation remaining quadratic and verification constant.
• The symbolic scheme scales to $\phi$5 states and thousands of obligations, at the cost of increased per-obligation overhead but state-space independence.

These results delineate a clear trade-off between scalability and system structure requirements. The explicit-state scheme supports arbitrary discrete structure but has polynomial scaling constraints. The symbolic scheme requires transition systems as guarded linear programs but can accommodate very large or infinite-state instances.

Theoretical and Practical Implications

The contribution of zero-knowledge model checking is significant for regulated domains that require safety and correctness certificates but also demand strong confidentiality guarantees. It enables new regulatory arrangements for software certification, standardization, and multi-party assessment where proprietary system exposition is not acceptable.

Theoretically, the work bridges proof-theoretic verification, automata-theoretic reasoning, and modern cryptographic ZK primitives. Critically, the treatment of proof obligations with both public and secret components, and their reduction to ZK satisfiability of efficiently checkable certificates, is novel. The integration of Farkas' lemma with zero-knowledge matrix algebra opens new avenues for cryptographically secure reasoning about symbolic verification tasks.

Limitations and Future Directions

The completeness of the approach is conditioned on the expressibility and existence of program certificates (ranking functions, Farkas witnesses) in the chosen encoding. For systems with complex or neural policies, or for non-linear or probabilistic specifications, further work on symbolic encoding and scalable ZK protocols for higher-order invariants will be necessary.

The framework inherits potential specification/certificate-induced leakage; certain specifications or certificates could, in principle, uniquely identify the system under analysis. Addressing this requires deeper research into the synthesis of non-leaking or privacy-preserving proof certificates.

Applying ZKMC to neural network verification, more general hybrid system verification, and supermartingale-based reasoning for probabilistic properties (cf. recent works such as "Neural Model Checking" [DBLP:conf/nips/GiacobbeKPT24] and "Complete Supermartingale Certificates for $\phi$6-Regular Properties" [lics26]) constitutes the most promising future directions.

Conclusion

This paper introduces a novel framework and practical schemes for certifying temporal properties of secret systems via public certificates and zero-knowledge discharging of verification obligations. By coupling automata-theoretic certification with advanced zero-knowledge and commitment primitives, the authors provide a strategic advancement with clear applications in confidential verification and regulated industry. This paradigm is poised to impact both the practice of software certification and the theoretical development of privacy-preserving formal methods.