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Printing Protocol: Physical ZKPs for Decomposition Puzzles

Suthee Ruangwises, Mitsugu Iwamoto

TL;DR

This work introduces a generic printing protocol that physically verifies solutions to decomposition puzzles via card-based zero-knowledge proofs. It constructs two concrete ZKP protocols, one for Five Cells and one for Meadows, by using templates corresponding to pentomino shapes and square sizes, respectively, and by coordinating chosen-cut, pile-shifting, and printing subprotocols. The schemes achieve perfect completeness, perfect soundness, and zero-knowledge, with resource usage scaling as $\Theta(mn)$ cards and shuffles for Five Cells and $\Theta(n^3)$ cards with related shuffles for Meadows. The results provide a practical, didactic approach to physical ZKPs for a broad class of grid-decomposition puzzles, while outlining limitations and directions for handling puzzles with larger (possibly exponential) template spaces.

Abstract

Decomposition puzzles are pencil-and-paper logic puzzles that involve partitioning a rectangular grid into several regions to satisfy certain rules. In this paper, we construct a generic card-based protocol called printing protocol, which can be used to physically verify solutions of decompositon puzzles. We apply the printing protocol to develop card-based zero-knowledge proof protocols for two such puzzles: Five Cells and Meadows. These protocols allow a prover to physically show that he/she knows solutions of the puzzles without revealing them.

Printing Protocol: Physical ZKPs for Decomposition Puzzles

TL;DR

This work introduces a generic printing protocol that physically verifies solutions to decomposition puzzles via card-based zero-knowledge proofs. It constructs two concrete ZKP protocols, one for Five Cells and one for Meadows, by using templates corresponding to pentomino shapes and square sizes, respectively, and by coordinating chosen-cut, pile-shifting, and printing subprotocols. The schemes achieve perfect completeness, perfect soundness, and zero-knowledge, with resource usage scaling as cards and shuffles for Five Cells and cards with related shuffles for Meadows. The results provide a practical, didactic approach to physical ZKPs for a broad class of grid-decomposition puzzles, while outlining limitations and directions for handling puzzles with larger (possibly exponential) template spaces.

Abstract

Decomposition puzzles are pencil-and-paper logic puzzles that involve partitioning a rectangular grid into several regions to satisfy certain rules. In this paper, we construct a generic card-based protocol called printing protocol, which can be used to physically verify solutions of decompositon puzzles. We apply the printing protocol to develop card-based zero-knowledge proof protocols for two such puzzles: Five Cells and Meadows. These protocols allow a prover to physically show that he/she knows solutions of the puzzles without revealing them.
Paper Structure (17 sections, 6 theorems, 7 figures)

This paper contains 17 sections, 6 theorems, 7 figures.

Table of Contents

  1. Introduction
  2. Five Cells
  3. Meadows
  4. Zero-Knowledge Proof
  5. Our Contribution
  6. Preliminaries
  7. Cards
  8. Pile-Shifting Shuffle
  9. Chosen Cut Protocol
  10. Printing Protocol
  11. ZKP Protocol for Five Cells
  12. Main Protocol
  13. Proof of Correctness and Security
  14. ZKP Protocol for Meadows
  15. Main Protocol
  16. ...and 2 more sections

Key Result

Lemma 1

If $P$ knows a solution of the Five Cells puzzle, then $V$ always accepts.

Figures (7)

  • Figure 1: An example of a $5 \times 5$ Five Cells puzzle (left) and its solution (right)
  • Figure 2: An example of a $7 \times 7$ Meadows puzzle (left) and its solution (right)
  • Figure 3: A pile-shifting shuffle on a $5 \times 6$ matrix
  • Figure 4: A $3 \times q$ matrix $M$ constructed in Step 1 of the chosen cut protocol
  • Figure 5: Examples of a succesful printing of numbers from a $4 \times 5$ template onto a $4 \times 5$ area (top) and an unsuccessful printing because of overlapping numbers (bottom)
  • ...and 2 more figures

Theorems & Definitions (12)

  • Lemma 1: Perfect Completeness
  • proof
  • Lemma 2: Perfect Soundness
  • proof
  • Lemma 3: Zero-Knowledge
  • proof
  • Lemma 4: Perfect Completeness
  • proof
  • Lemma 5: Perfect Soundness
  • proof
  • ...and 2 more