NP-Completeness and Physical Zero-Knowledge Proofs for Zeiger

TL;DR

This paper proves that solving Zeiger puzzles is NP-complete and introduces a physical zero-knowledge proof protocol allowing one to demonstrate a solution's existence without revealing it.

Contribution

It establishes NP-completeness of Zeiger puzzles and develops a novel physical zero-knowledge proof protocol for them.

Findings

NP-completeness of Zeiger decision problem

Construction of a physical zero-knowledge proof protocol

Feasibility of physically demonstrating solutions without revealing them

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.