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NP-Completeness and Physical Zero-Knowledge Proofs for Zeiger

Suthee Ruangwises

TL;DR

This work establishes the computational hardness of Zeiger by proving its solvability decision problem is NP-complete through a polynomial-time reduction from $\text{NAE3SAT}^+$. It then delivers a card-based physical zero-knowledge proof protocol for Zeiger, built from a suite of card-encoded primitives (copy, set size, summation, and comparing) and an encoding scheme with parameter $b=\max\{k,\ell\}$ that ensures perfect completeness, perfect soundness, and zero-knowledge. The main protocol encodes numbers on the grid as $E_b(x)$s and verifies, for each cell, that the number of distinct values among its directional neighbors matches the cell's value, using $\Theta(bk\ell)$ cards and shuffles. This yields a practically interpretable, computer-free ZKP for Zeiger, with implications for understanding the complexity and verifiability of pencil puzzles and for educational demonstrations of ZKPs.

Abstract

Zeiger is a pencil puzzle consisting of a rectangular grid, with each cell having an arrow pointing in horizontal or vertical direction. Some cells also contain a positive integer. The objective of this puzzle is to fill a positive integer into every unnumbered cell such that the integer in each cell is equal to the number of different integers in all cells along the direction an arrow in that cell points to. In this paper, we prove that deciding solvability of a given Zeiger puzzle is NP-complete via a reduction from the not-all-equal positive 3SAT (NAE3SAT+) problem. We also construct a card-based physical zero-knowledge proof protocol for Zeiger, which enables a prover to physically show a verifier the existence of the puzzle's solution without revealing it.

NP-Completeness and Physical Zero-Knowledge Proofs for Zeiger

TL;DR

This work establishes the computational hardness of Zeiger by proving its solvability decision problem is NP-complete through a polynomial-time reduction from . It then delivers a card-based physical zero-knowledge proof protocol for Zeiger, built from a suite of card-encoded primitives (copy, set size, summation, and comparing) and an encoding scheme with parameter that ensures perfect completeness, perfect soundness, and zero-knowledge. The main protocol encodes numbers on the grid as s and verifies, for each cell, that the number of distinct values among its directional neighbors matches the cell's value, using cards and shuffles. This yields a practically interpretable, computer-free ZKP for Zeiger, with implications for understanding the complexity and verifiability of pencil puzzles and for educational demonstrations of ZKPs.

Abstract

Zeiger is a pencil puzzle consisting of a rectangular grid, with each cell having an arrow pointing in horizontal or vertical direction. Some cells also contain a positive integer. The objective of this puzzle is to fill a positive integer into every unnumbered cell such that the integer in each cell is equal to the number of different integers in all cells along the direction an arrow in that cell points to. In this paper, we prove that deciding solvability of a given Zeiger puzzle is NP-complete via a reduction from the not-all-equal positive 3SAT (NAE3SAT+) problem. We also construct a card-based physical zero-knowledge proof protocol for Zeiger, which enables a prover to physically show a verifier the existence of the puzzle's solution without revealing it.
Paper Structure (17 sections, 6 theorems, 4 figures, 4 algorithms)

This paper contains 17 sections, 6 theorems, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Zero-Knowledge Proof
  3. Physical Zero-Knowledge Proof Protocols
  4. Our Contribution
  5. NP-Completeness Proof of Zeiger
  6. Physical ZKP Protocol for Zeiger
  7. Preliminaries
  8. Cards
  9. Pile-Shifting Shuffle
  10. Pile-Scramble Shuffle
  11. Copy Protocol
  12. Set Size Protocol
  13. Summation Protocol
  14. Comparing Protocol
  15. Our Main Protocol
  16. ...and 2 more sections

Key Result

Lemma 1

For each $q \leq n$, there are exactly two ways to fill numbers into all unnumbered cells in the $q$-th column of $G$ to satisfy constraints of all up and down arrows in that column, which are filling all 2s and filling all 3s.

Figures (4)

  • Figure 1: An example of a $5 \times 5$ Zeiger puzzle (left) and its solution (right)
  • Figure 2: An NAE3SAT+ instance (left) and one of its possible solutions (right)
  • Figure 3: A $7 \times 10$ Zeiger puzzle grid transformed from the NAE3SAT+ instance in Fig. \ref{['fig2']}
  • Figure 4: A solution of the Zeiger puzzle in Fig. \ref{['fig3']} transformed from the NAE3SAT+ solution in Fig. \ref{['fig2']}

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3: Perfect Completeness
  • proof
  • Lemma 4: Perfect Soundness
  • proof
  • ...and 2 more