NP-Completeness and Physical Zero-Knowledge Proofs for Zeiger

2024-09-22

arXiv.org

AI Summary

This work investigates the computational complexity and physical zero-knowledge verifiability of the Zeiger pencil puzzle. First, it establishes—via a polynomial-time reduction from NAE3SAT+—that solving the Zeiger puzzle is NP-complete, resolving its computational hardness for the first time. Second, it proposes the first physical zero-knowledge proof protocol using standard playing cards, executable manually without electronic devices, and rigorously satisfying completeness, zero-knowledge, and honest-verifier soundness. The protocol reveals no information about the solution while enabling physical verification of solution existence. By bridging combinatorial logic modeling and physical cryptography, this work establishes a novel connection between theoretical complexity analysis and practical privacy-preserving verification, providing the first formal framework and implementable solution for trustworthy physical verification of tangible puzzles.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Responsible Web: Data and user privacy-enhancing technologies for the Web

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.

Problem

• Proving Zeiger puzzle's NP-completeness
• Reduction from NAE3SAT+ problem
• Developing card-based zero-knowledge proof protocol

Innovation

• NP-completeness proof
• card-based protocol
• zero-knowledge proof