Zero-Knowledge Position Verification

• Zero-Knowledge Position Verification is a cryptographic protocol that validates spatial claims while preserving confidentiality by concealing detailed location information.
• It employs commitment schemes, encryption, and NP zero-knowledge proofs to prevent spoofing and enforce spatial soundness in both quantum and classical settings.
• The protocol underpins privacy-preserving applications in vehicular access, secure robotics, and authenticated data structures by combining cryptographic assurances with physical position constraints.

Zero-knowledge position verification (ZK-PV) refers to cryptographic protocols in which an agent proves statements about its position (typically spatial or spatiotemporal) such that the verifier is convinced of the statement’s truth but gains no knowledge beyond validity. This class of protocols has become foundational in privacy-preserving location access control, secure collaborative robotics, anonymous order queries in authenticated data structures, and the broader study of position-based cryptography. ZK-PV subsumes standard position-verification by combining spatial soundness (protection against colluding adversaries spoofing their coordinates) with formal zero-knowledge privacy—revealing nothing about the true position except what is logically implied by the claim itself (Girish et al., 26 Jan 2026, Li et al., 2023, Ghosh et al., 2014).

1. Formal Definitions and Security Properties

Let $R \subseteq \mathbb{R}^d \times \mathbb{R}$ be a set of allowed spacetime points, with a family of honest-prover algorithms $\{P_\alpha\}_{\alpha \in R}$ situated at point $\alpha = (L, t)$. A position-verification (PV) protocol $(P, V, R)$ has completeness error $c(\kappa)$ and position security $s(\kappa)$ against spoofing coalitions $\mathcal{C}$ if:

• Completeness: For all $\alpha \in R$, the probability that $(P_\alpha \leftrightarrow V)(1^\kappa)$ is accepted is at least $1 - c(\kappa)$.
• Position Security: For any coalition $P^* \in \mathcal{C}$ not occupying any $\alpha \in R$, the probability of acceptance is at most $s(\kappa)$.

A ZK-PV protocol additionally ensures (honest-verifier) computational zero-knowledge: $$\{\text{Sim}(1^\kappa, \tau)\}_\kappa \approx_c \{\text{View}_V^\tau(P_\alpha \leftrightarrow V)(1^\kappa)\}_\kappa$$ where $\text{Sim}$ is a quantum or classical simulator, and View denotes the verifier’s view at time $\tau$ (Girish et al., 26 Jan 2026).

The core security criteria are:

• Completeness: Honest provers at valid positions or for valid position claims always pass.
• Soundness: Cheating provers can only succeed by breaking strong cryptographic (e.g., BLS, commitment) assumptions or underlying physical-spatial constraints.
• Zero-Knowledge: No information except for the truth of the statement is leaked; transcripts are simulatable given only public claims.

2. Position Commitments and Zero-Knowledge Constructions

Central to modular ZK-PV constructions is the position commitment (PC) primitive (Girish et al., 26 Jan 2026):

• A PC scheme for a finite $S \subseteq \mathbb{R}^d \times \mathbb{R}$ is a protocol $(\text{Commit}, \text{Reveal})$ such that after a Commit phase (up to $t_\text{final}$), verifiers possess a transcript-state $\rho$. In Reveal, the prover opens at position $\alpha^*$; verifiers accept if and only if protocol correctness is satisfied.

The encrypt-then-verify construction uses:

• A “nice” singleton PV protocol family $\{\Pi_\alpha\}_{\alpha \in S}$.
• Post-quantum secret-key encryption $(\text{Gen, Enc, Dec})$ and statistically binding, computationally hiding commitment $(\text{Setup, Com})$.

Commit: Prover encrypts PV responses at a single position (honest) and simulates dummy encryptions elsewhere, hiding the real position. Reveal: Decrypts only the honest position’s transcript for verification.

The resulting ZK-PV protocol for claims $R \subseteq S$ is obtained via a classical NP zero-knowledge proof that the opening (the $\rho$ commitment) is valid for some $\alpha \in R$. This achieves honest-verifier zero-knowledge and statistical soundness bounded by $|S| \cdot s(\kappa)$ (Girish et al., 26 Jan 2026).

3. Cryptographic Building Blocks in Spatial ZK Proofs

Pairing-based cryptography, homomorphic commitments, and zero-knowledge set constructions are foundational in applied ZK-PV (Li et al., 2023, Ghosh et al., 2014):

• Pairing Groups: Let $G, G_T$ be prime order cyclic groups (e.g., BLS12-381), $g \in G$. A bilinear pairing $e : G \times G \to G_T$ is used for short, pairing-based ZK proofs.
• Commitment Schemes: Hash-to-curve functions $H:\{0,1\}^* \to G$ (e.g., SHA-256 plus map-to-curve) are used to derive group element commitments.
• Zero-Knowledge List/Order Proofs: For totally ordered sets, protocols such as Zero-Knowledge Lists (ZKL) and Privacy-Preserving Authenticated Lists (PPAL) use homomorphic integer commitments, aggregate signatures, and non-interactive ZK proofs of non-negativity to attest to membership and relative orderings on a list in ZK (Ghosh et al., 2014).

4. Protocol Workflows and Concrete Implementations

Quantum Model Protocols

Quantum protocols permit position-secure ZK constructions infeasible in the classical model, leveraging limits on entanglement and signal propagation. In (Girish et al., 26 Jan 2026), the ZK-PV workflow proceeds as:

1. Commit: Verifiers broadcast challenge signals over all $\alpha \in S$. The honest prover, situated at $\alpha^*$, replies in a protocol execution only at that point, encrypting its measurement outcome; elsewhere, simulated dummy encryptions are sent.
2. Reveal: The prover opens its commitment by revealing secret key and randomness, verifiers decrypt only the matching responses, and verify position claims via underlying PV predicates.
3. Zero-Knowledge Proof: An NP proof (e.g., post-quantum ZK) that the opening admits a valid transcript for some position in $R$, hiding all information except the logical content of the statement.

Classical and Blockchain-Based Protocols

In dynamic vehicular platooning, ZK position (and identity) verification on permissioned blockchains is realized as follows (Li et al., 2023):

• Entities: Trusted permission issuer, autonomous trucks as provers (unique secret key $a_i$), a geofenced subset of local verifiers, and a Hyperledger Fabric blockchain.
• Protocol: The truck $i$ forms a proof $\delta_i = h_i^{a_i}$ with $h_i = H(m_i)$, and broadcasts $(\delta_i, h_i)$ to verifiers; each verifier checks the pairing equation $e(\delta_i, g) = e(h_i, v_i)$; results are aggregated and acted upon in the blockchain.
• Spatial Scope: Only physically co-located, authenticated trucks are eligible as verifiers, enforcing locality by policy not by cryptographic range proofs.

PPAL schemes for zero-knowledge order queries in lists use aggregation and witness techniques to ensure that the client learns nothing beyond the membership and order of queried items (Ghosh et al., 2014).

5. Security Analysis and Efficiency

Theoretical and empirical analyses establish security under standard cryptographic assumptions and quantify trade-offs (Girish et al., 26 Jan 2026, Li et al., 2023, Ghosh et al., 2014):

• Soundness: Relies on position security of the underlying protocol (in quantum PV), binding of commitments, and difficulty of discrete log, n-BDHI, or BLS forgery.
• Zero-Knowledge: Achieved through encryption and commitment hiding, and post-quantum NP ZK proofs in modular frameworks.
• Performance:
  • In BLS pairing-based schemes (ZK identity/position proof): proof size ≈ 96 bytes, generation ≈ 29 ms, verification per peer ≈ 210 ms (Li et al., 2023).
  • Hyperledger Fabric: up to 27 tps under minimal endorsement, join latency drops ≈ 30% with geofenced/local verification, ZK proof overhead scales $O(k)$ with verifiers.
  • In the list order ZKL/PPAL setting: ZKL has $O(mk)$ proof size and time, PPAL reduces to $O(m)$-size proofs and $O(m\log n)$ server time with pairings (Ghosh et al., 2014).
  • Quantum ZK-PV: Communication in the general construction is $O(|S| \cdot \text{poly}(\kappa))$; with broadcast-only protocols, per-time-step cost drops to $O(1)$.

Comparative table of key scheme properties:

Scheme	Security Model	ZK Property	Efficiency
(Girish et al., 26 Jan 2026) Quantum ZK-PV	Quantum (QPT, entanglement-bound)	Honest-verifier, post-quantum	General: $O(|S|)$ comm.; broadcast: $O(1)$/step
(Li et al., 2023) BLS Platooning	Classical, blockchain	Computational (ZK of key)	Proof $\sim$96 bytes, $<$250ms verify
(Ghosh et al., 2014) ZKL/PPAL	Classical, list order	Statistical/comp. ZK, client	ZKL: $O(mk)$, PPAL: $O(m)$; pairing-based

6. Applications and Extensions

ZK-PV finds application in multiple domains:

• Vehicular and IoT Access Control: Privacy-preserving geofenced authentication of vehicles or devices to physical zones or platoons (Li et al., 2023).
• Spacetime Statements: Proving complex claims ("I was not near $L$ at $t$") with strong privacy (Girish et al., 26 Jan 2026).
• Authenticated Data Structures: ZK queries for membership and sequential position in dynamic lists, enabling privacy-preserving query services over cloud-stored data (Ghosh et al., 2014).

Extensions include modular addition of range proofs for continuous location claims, Bulletproof-like range ZK (future direction in (Li et al., 2023)), and adaptation to more expressive spatial or temporal predicates.

7. Trade-offs, Limitations, and Future Directions

The design and deployment of ZK-PV schemes reflect trade-offs between efficiency, soundness amplification, and locality:

• Protocols leveraging physical-layer trust (e.g., DSRC, authenticated geofencing) avoid explicit cryptographic range proofs, but rely on environmental assumptions (Li et al., 2023).
• The general ZK-PV construction in the quantum setting exhibits linear communication and computation with the size of the region $|S|$, but mesh or broadcast-optimized versions mitigate this.
• Increasing endorsement thresholds in blockchain-based protocols improves security at the expense of throughput and latency.
• The feasibility of unconditionally secure classical position-verification is ruled out; quantum effects and bound entanglement are necessary for strong position security (Girish et al., 26 Jan 2026).
• Future work may focus on integrating structured range proofs into modular ZK-PV, enhanced resistance to entangled spoofing coalitions, or scalable order/position query proofs for large-scale data (Li et al., 2023, Girish et al., 26 Jan 2026, Ghosh et al., 2014).

Key research cited: "Private Proofs of When and Where" (Girish et al., 26 Jan 2026), "Location-aware Verification for Autonomous Truck Platooning Based on Blockchain and Zero-knowledge Proof" (Li et al., 2023), and "Verifiable Member and Order Queries on a List in Zero-Knowledge" (Ghosh et al., 2014).