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Verifying the First Nonzero Term: Physical ZKPs for ABC End View, Goishi Hiroi, and Toichika

Suthee Ruangwises

TL;DR

This work introduces a physical, card-based protocol to verify the first nonzero term of a sequence with $Θ(1)$ shuffles and a single-card encoding per term, improving over prior $Θ(n)$-shuffle approaches. The core FirstNonZero protocol is then leveraged to construct ZKP procedures for three classic pencil-and-paper puzzles—ABC End View, Goishi Hiroi, and Toichika—demonstrating perfect completeness, perfect soundness, and zero-knowledge. The ABC End View protocol uses a permutation-verification primitive to enforce row/column uniqueness, while Goishi Hiroi and Toichika extend the idea to sequential stone-picking and region-arrow constraints with specialized encodings and non-adjacency checks. Collectively, the results provide a practical, intuition-friendly framework for verifiable, non-revealing puzzle solving using only physical cards, with potential implications for teaching and demonstration of ZKPs in tangible settings.

Abstract

In this paper, we propose a physical protocol to verify the first nonzero term of a sequence using a deck of cards. The protocol lets a prover show the value of the first nonzero term of a given sequence to a verifier without revealing which term it is. Our protocol uses $Θ(1)$ shuffles, which is asymptotically lower than that of an existing protocol of Fukusawa and Manabe which uses $Θ(n)$ shuffles, where $n$ is the length of the sequence. We also apply our protocol to construct zero-knowledge proof protocols for three well-known logic puzzles: ABC End View, Goishi Hiroi, and Toichika. These protocols enables a prover to physically show that he/she know solutions of the puzzles without revealing them.

Verifying the First Nonzero Term: Physical ZKPs for ABC End View, Goishi Hiroi, and Toichika

TL;DR

This work introduces a physical, card-based protocol to verify the first nonzero term of a sequence with shuffles and a single-card encoding per term, improving over prior -shuffle approaches. The core FirstNonZero protocol is then leveraged to construct ZKP procedures for three classic pencil-and-paper puzzles—ABC End View, Goishi Hiroi, and Toichika—demonstrating perfect completeness, perfect soundness, and zero-knowledge. The ABC End View protocol uses a permutation-verification primitive to enforce row/column uniqueness, while Goishi Hiroi and Toichika extend the idea to sequential stone-picking and region-arrow constraints with specialized encodings and non-adjacency checks. Collectively, the results provide a practical, intuition-friendly framework for verifiable, non-revealing puzzle solving using only physical cards, with potential implications for teaching and demonstration of ZKPs in tangible settings.

Abstract

In this paper, we propose a physical protocol to verify the first nonzero term of a sequence using a deck of cards. The protocol lets a prover show the value of the first nonzero term of a given sequence to a verifier without revealing which term it is. Our protocol uses shuffles, which is asymptotically lower than that of an existing protocol of Fukusawa and Manabe which uses shuffles, where is the length of the sequence. We also apply our protocol to construct zero-knowledge proof protocols for three well-known logic puzzles: ABC End View, Goishi Hiroi, and Toichika. These protocols enables a prover to physically show that he/she know solutions of the puzzles without revealing them.
Paper Structure (27 sections, 6 theorems, 1 equation, 7 figures)

This paper contains 27 sections, 6 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Protocol of Fukusawa and Manabe
  4. Our Contribution
  5. Preliminaries
  6. Cards
  7. Pile-Shifting Shuffle
  8. Chosen Cut Protocol
  9. Verifying the First Nonzero Term of a Sequence
  10. ZKP Protocol for ABC End View
  11. Uniqueness Verification Protocol
  12. ZKP Protocol for Goishi Hiroi
  13. Idea of the Protocol
  14. Setup
  15. Main Protocol
  16. ...and 12 more sections

Key Result

Lemma 1

If $P$ knows a solution of the Goishi Hiroi puzzle, then $V$ always accepts.

Figures (7)

  • Figure 1: An example of a pile-shifting shuffle on a $4 \times 5$ matrix
  • Figure 2: A $3 \times n$ matrix $M$ constructed in Step 1 of the chosen cut protocol
  • Figure 3: An example of a $5 \times 5$ ABC End View puzzle with letters from the range A, B, and C (left) and its solution (right)
  • Figure 4: A $2 \times q$ matrix constructed in Step 1
  • Figure 5: An example of a $6 \times 6$ Goishi Hiroi puzzle with 12 stones (left) and its solution with each number $i$ indicating the $i$-th stone that was picked (right)
  • ...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