Zero-Knowledge Spatial Checks

• Zero-Knowledge Spatial Checks are protocols that verify geometric relations without disclosing specific information, using both relativistic and algebraic techniques.
• They employ efficient SNARK-compatible circuit representations and physical separation constraints to ensure information-theoretic security in practical applications.
• These methods integrate digital geometry with multi-prover interactive proofs and even physical card-based systems to address scalability and privacy challenges.

Zero-knowledge spatial checks are cryptographic protocols that allow a prover to demonstrate knowledge of a geometric/spatial relation—such as occupying a region, maintaining spatial separation, or possessing a solution to a spatial constraint—without revealing any information beyond the validity of the claim itself. These protocols leverage spatial structure at both the algebraic and physical levels, employing principles from special relativity (relativistic no-signalling), digital geometry, and efficient circuit representations. They constitute the foundation of recent advances in privacy-preserving location proofs, physical cryptography, and practical relativistic zero-knowledge for NP-complete problems.

1. Foundations and Physical Principles

Zero-knowledge spatial checks exploit the geometry of space and time in two major ways: (a) enforcing non-communication via physical separation (relativistic protocols), and (b) encoding and verifying spatial/geometric predicates in zero-knowledge proofs (ZKPs).

Relativistic protocols use the postulate that no signal can travel faster than light ($c$); by placing multiple provers far apart and bounding the response time to $\Delta t < d/c$ (where $d$ is the distance between provers), the protocol enforces a no-signalling constraint: the responses from different provers cannot depend on each other's challenges in any round (Alikhani et al., 2020, Crépeau et al., 2019). This yields information-theoretic security based on the laws of physics, eliminating reliance on computational assumptions such as one-way functions.

Geometric/Algebraic protocols encode spatial predicates—such as point-in-region, proximity, path connectivity, or shape constraints—into arithmetic circuits or card-based encodings, making them amenable to implementation in non-interactive SNARKs, interactive ZK proofs, or even physical cryptography with standard decks of cards (Ernstberger et al., 2024, Bogdanov et al., 20 Jun 2025, Lee et al., 30 Dec 2025, Hart et al., 2022, Ruangwises et al., 2022).

2. Relativistic Zero-Knowledge Proofs: Protocols and Security

Relativistic ZK protocols for NP (notably 3-coloring and NP-complete variants) follow a multi-prover interactive proof (MIP) architecture, where each prover receives their challenge simultaneously and must respond within a tight bound, precluding any communication during the round.

• Spatial configuration: Two (or more) prover-verifier pairs (P₁,V₁), (P₂,V₂) are placed at a known separation ($d$), and the entire round—challenge issuance to response receipt—must complete within $\Delta t < d/c$ (Alikhani et al., 2020). For instance, $d=60$ m yields $\Delta t \approx 200$ ns; practical systems use GPS-disciplined FPGAs and laser-synchronized triggers.
• Protocol phases: In the case of 3-coloring:
  1. Both provers pre-share a random 3-coloring and appropriate secret labellings.
  2. Each verifier independently selects a challenge (random edge, masking bit) and transmits to the local prover.
  3. Provers reply with masked colour data. Verifiers collect responses and perform one of two spatially-enforced checks: the colour test verifies proper coloring, while the consistency test enforces non-adaptive well-definition.

• Security: The relativistic timing constraint enforces a classical no-signalling condition: $P_1$'s answer cannot depend on $P_2$'s challenge (and vice versa). This replaces the cryptographic binding property of computational commitments with a physical one (Crépeau et al., 2019). For completeness, honest provers always succeed; for soundness, any cheating pair is caught with exponentially small failure except (even against entangled provers, with an extra consistency test).

Table: Sample Parameter Regimes from (Alikhani et al., 2020)

Setup	Separation $d$	Time Bound $\Delta t$	Rounds (k=100)	Total Time
Short-distance	60 m	192 ns	$\sim$548,500	$\sim$1 s
Long-distance	390 m	840 ns	$\sim$548,500	$\sim$2 s

This paradigm achieves post-quantum, information-theoretic zero-knowledge for all NP with practical scalability (graphs with $\sim10^4$ edges in seconds), minimal hardware (standard FPGAs and optics), and (unlike classical cut-and-choose) remains secure against advances in quantum cryptanalysis (Alikhani et al., 2020, Crépeau et al., 2019).

3. Spatial Zero-Knowledge with Digital Geometry

Zero-knowledge spatial checks can be instantiated as privacy-preserving proofs for location and region membership without the need for spatially-separated provers. These protocols utilize efficient representation and SNARK-friendly circuit encodings for geometric relations, supporting use cases from geofencing to location-blind digital credentials.

• ZK Location Privacy: Protocols such as ZKLP enable users to prove knowledge of coordinates $(\lambda_v,\phi_v)$ inside some region, e.g., a convex polygon or a circle, without revealing location. Region membership checks (point-in-polygon, within-distance) are compiled into efficient SNARK circuits using custom IEEE-754 floating-point arithmetic, amortizing constraint cost over circuit size (Ernstberger et al., 2024). Optimizations leverage hints, branchless geometric reductions, and lookup amortization, yielding sub-300 ms proving times for nontrivial predicates.
• Trajectory and Territorial Proofs: Proof-of-location for vehicle taxation or subsidies involves expressing predicates such as “trajectory remained within region $R$ for percentage $P_{req}$ of the time” or “never left a union of polygons.” Polygonal and circular inclusion is achieved via barycentric coordinate checks and in-circuit range proofs (Bogdanov et al., 20 Jun 2025).

STARK-based zone checks implement scalable, efficient PiP (Point-in-Polygon) checks through grid-based lookup tables. Boolean grid encodings provide a coarse indicator of inclusion, while distance-aware grid encodings quantize signed distance to the zone boundary and interpolate in-circuit. This increases accuracy at low grid resolutions, with only a moderate increase in verification time and proof size (Lee et al., 30 Dec 2025).

Table: Comparison of Grid Encodings for PiP (Lee et al., 30 Dec 2025)

Scheme	Accuracy (r=16)	Proof Size	Prove/Verify Time
Boolean center	45%	120 kB	0.50 s / 20 ms
Distance-aware	61%	168 kB	0.60 s / 28 ms

4. Physical ZKPs: Card-Based Spatial Proofs

Spatial zero-knowledge checks can be realized physically, leveraging card decks and spatially-arranged encodings as one-time pads and secret indicators. These constructions achieve perfect (information-theoretic) zero-knowledge and soundness using basic combinatorial arguments.

• Graph and Path Constraints: For NP-hard spatial problems such as Hamiltonian cycle or k-disjoint covering paths, face-down stacks of cards encode solutions as “one-hot” indicators and path/order labels. Verification proceeds via local spatially-constrained checks (such as degree 2 at each vertex, label adjacency modulo $n$), implemented by randomized matrix shuffles and partial reveals. Card-based spatial checks enforce all adjacency and covering constraints without leaking any information about the solution (Hart et al., 2022).
• Grid Region Verification: The Ruangwises–Itoh “flooding” protocol verifies axis-aligned rectangles (as in Shikaku) in an $m \times n$ grid: a sequence of two “floods” (secret propagation of marker colors) proves area and rectangularity, enforced by local consistent flipping and secret direction-encodings. The only observable data are global statistical properties, independent of the rectangle location (Ruangwises et al., 2022).

These physical protocols are concrete realizations of spatial zero-knowledge, requiring only $O(mn)$ cards and generic shuffling procedures, and are naturally immune to computational or quantum attacks.

5. Relativistic and Algebraic MIP Zero-Knowledge: General Theory

The unifying abstraction for zero-knowledge spatial checks is the Multi-Prover Interactive Proof (MIP) model, augmented by explicit correlator classes describing the non-local resources: $\mathbb{LOC}$ (local randomness), $\mathbb{ENT}$ (shared entanglement), $\mathbb{NOSIG}$ (no-signalling), $\mathbb{SIG}$ (full signalling) (Crépeau et al., 2019, Chiesa et al., 2018).

• No-signalling as a physical check: By imposing $\Delta t < d/c$, one guarantees that the honest provers are local/non-signalling, enforcing soundness by relativity and zero-knowledge by protocol design (Alikhani et al., 2020, Crépeau et al., 2019, Crépeau et al., 2019).
• Algebraic MIPs for NEXP: Spatially separated (even entangled) provers made possible to achieve perfect zero-knowledge for all NEXP by lifting zero-knowledge interactive PCPs via low-degree tests and algebraic masking. These protocols are sound against quantum entangled cheating provers (in the MIP* model) when spatial separation is enforced, and simulate transcript distributions for any malicious verifier (Chiesa et al., 2018).

LE-MIP frameworks characterize exactly the non-local “advantage” required by simulators, with careful protocol design realizing zero-knowledge with only PR-boxes (no-signalling resources), while maintaining soundness for less potent adversaries (Crépeau et al., 2019).

6. Implementation, Performance, and Applications

Zero-knowledge spatial checks are now practical across hardware and context scales:

• Hardware-level realization: Off-the-shelf FPGAs and standard optical interconnects suffice to implement relativistic ZKPs with sub-microsecond timing, with scaling to $|V| \sim 10^4$ and $|E| \sim 10^4$ possible in seconds, and even better with ASICs (Alikhani et al., 2020).
• Software protocols: SNARK-based region proofs for single-point or trajectory containment produce short proofs (192 B - 168 kB), with prover times down to hundreds of ms and online verification rates in the hundreds to low thousands per second, well within real-time thresholds for geofencing or location credentials (Ernstberger et al., 2024, Bogdanov et al., 20 Jun 2025, Lee et al., 30 Dec 2025).
• Physical proofs: Card-based spatial checks are immediate to realize and require only linear resources in the input size, with no computational hardness assumptions (Hart et al., 2022, Ruangwises et al., 2022).
• Applications: Zero-knowledge spatial checks are deployed or prototyped for identification systems (e.g., bank ATMs with dual-chip provers), privacy-preserving geofencing, location-based regulatory compliance (EV subsidies, tax zones), blockchain and smart contracts, secure voting/auctions, nuclear verification, and puzzle verification (Alikhani et al., 2020, Bogdanov et al., 20 Jun 2025, Lee et al., 30 Dec 2025, Hart et al., 2022).

7. Open Challenges and Future Directions

Significant theoretical and practical challenges remain:

• Quantum adversaries: Extending information-theoretic soundness to protocols secure against quantum adversaries (MIP*) with only two provers without exponential overhead is open; all known three-prover or graph inflation approaches are currently impractical for real-world parameters (Alikhani et al., 2020, Chiesa et al., 2018).
• Hardware/time optimization: Achieving sub-10 ns round-trip times and millimeter spatial separations with ASICs and parallel I/O could miniaturize relativistic protocols further (Alikhani et al., 2020).
• Complex spatial relations: Efficient, succinct ZKPs for spatial properties beyond point-in-region—such as path homotopy, connectivity, or high-dimensional constraints—remain a rich research area (Bogdanov et al., 20 Jun 2025, Lee et al., 30 Dec 2025).
• Aggregated proofs: Roll-up and aggregation of many spatial proofs, especially for high-throughput regulatory and financial applications, is an ongoing systems challenge (Bogdanov et al., 20 Jun 2025).

Recent developments demonstrate that by combining physical spatial separation, efficient algebraic encoding, and tight hardware integration, zero-knowledge spatial checks achieve a unique blend of information-theoretic, post-quantum, and practical privacy guarantees. These methods continue to expand the frontiers of both theoretical cryptography and applied privacy technologies.