Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spatial Discretization for Fine-Grain Zone Checks with STARKs

Sungmin Lee, Kichang Lee, Gyeongmin Han, JeongGil Ko

TL;DR

This work tackles zero-knowledge spatial zone checks (PiP) by discretizing space into grid-based lookup tables within a fixed STARK execution model. It compares center-point, voting-based boolean, and distance-aware discretizations, demonstrating that storing signed distance values with arithmetic-native bilinear interpolation yields substantial accuracy gains on coarse grids with only modest proof overhead. The study provides a systematic evaluation on real OpenStreetMap zones, identifying that accuracy saturates around moderate resolutions while proof costs scale with grid size, and that distance-aware encodings shift the trade-off favorably. The findings highlight discretization strategy, particularly distance-aware encodings, as a key lever for efficient, private spatial checks in practice, enabling scalable zero-knowledge geospatial applications.

Abstract

Many location-based services rely on a point-in-polygon test (PiP), checking whether a point or a trajectory lies inside a geographic zone. Since geometric operations are expensive in zero-knowledge proofs, privately performing the PiP test is challenging. In this paper, we answer the research questions of how different ways of encoding zones affect accuracy and proof cost by exploiting gridbased lookup tables under a fixed STARK execution model. Beyond a Boolean grid-based baseline that marks cells as in- or outside, we explore a distance-aware encoding approach that stores how far each cell is from a zone boundary and uses interpolation to reason within a cell. Our experiments on real-world data demonstrate that the proposed distance-aware approach achieves higher accuracy on coarse grids (max. 60%p accuracy gain) with only a moderate verification overhead (approximately 1.4x), making zone encoding the key lever for efficient zero-knowledge spatial checks.

Spatial Discretization for Fine-Grain Zone Checks with STARKs

TL;DR

This work tackles zero-knowledge spatial zone checks (PiP) by discretizing space into grid-based lookup tables within a fixed STARK execution model. It compares center-point, voting-based boolean, and distance-aware discretizations, demonstrating that storing signed distance values with arithmetic-native bilinear interpolation yields substantial accuracy gains on coarse grids with only modest proof overhead. The study provides a systematic evaluation on real OpenStreetMap zones, identifying that accuracy saturates around moderate resolutions while proof costs scale with grid size, and that distance-aware encodings shift the trade-off favorably. The findings highlight discretization strategy, particularly distance-aware encodings, as a key lever for efficient, private spatial checks in practice, enabling scalable zero-knowledge geospatial applications.

Abstract

Many location-based services rely on a point-in-polygon test (PiP), checking whether a point or a trajectory lies inside a geographic zone. Since geometric operations are expensive in zero-knowledge proofs, privately performing the PiP test is challenging. In this paper, we answer the research questions of how different ways of encoding zones affect accuracy and proof cost by exploiting gridbased lookup tables under a fixed STARK execution model. Beyond a Boolean grid-based baseline that marks cells as in- or outside, we explore a distance-aware encoding approach that stores how far each cell is from a zone boundary and uses interpolation to reason within a cell. Our experiments on real-world data demonstrate that the proposed distance-aware approach achieves higher accuracy on coarse grids (max. 60%p accuracy gain) with only a moderate verification overhead (approximately 1.4x), making zone encoding the key lever for efficient zero-knowledge spatial checks.
Paper Structure (20 sections, 2 equations, 3 figures)

This paper contains 20 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. STARK and Lookup-Based Computation
  4. Spatial Predicates and Discretization under Zero Knowledge
  5. Methodology
  6. Problem Setting
  7. Lookup-Based Discretization (RQ1)
  8. Grid-Based Lookup Model
  9. Center-Point Discretization (Boolean)
  10. Voting-Based Discretization (Boolean)
  11. Distance-Aware Discretization (SDF + Interpolation)
  12. Evaluation
  13. Experimental Setup
  14. Datasets
  15. Metrics
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Comparison of lookup-based discretization strategies. The color gradient in panel (c) visualizes continuous boundary proximity recovered via bilinear interpolation.
  • Figure 2: Spatial distribution of semantic zone categories of our dataset. Colors indicate representative zone types for visualization, while zones may overlap in practice. The figure highlights heterogeneous geometries, including compact buildings and narrow road segments.
  • Figure 3: Accuracy and proof-cost trends across grid resolutions and discretization strategies. For panel (a), (b), and (c), lines and markers correspond to center-point (red triangles), voting-based (blue squares), and distance-aware (green circles) discretization. In panel (d), color encodes grid resolution: darker purple indicates coarser grids (from $2\times2$), while brighter green corresponds to finer grids (up to $128\times128$).