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Card-Based ZKP Protocols for Connectivity-Based Puzzles: Extending to Tree Structures with Application to Nurimeizu

Authors Daiki Miyahara , Pascal Lafourcade , Maxime Puys

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LIPIcs.FUN.2026.35.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic protocols
Keywords
  • Card-based cryptography
  • Nurimeizu
  • Nikoli
  • ZKP

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Abstract

Card-based zero-knowledge proof (ZKP) protocols allow a prover to convince a verifier that it knows a witness of a given statement, without revealing any information, using a physical deck of playing cards. Previous studies have focused on puzzles with a specific connected component, such as a simple cycle and a polyomino. In this study, we propose a unified approach to handle a family of connected components, including a tree, path, cycle, and polyomino. This approach achieves this verification in O(mn) steps relative to a given grid size m × n. Using this approach, we construct a card-based ZKP protocol for Nurimeizu, where the goal is to find the shortest path on a given grid.

Cite As Get BibTex

Daiki Miyahara, Pascal Lafourcade, and Maxime Puys. Card-Based ZKP Protocols for Connectivity-Based Puzzles: Extending to Tree Structures with Application to Nurimeizu. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.35

Author Details

Daiki Miyahara
  • The University of Electro-Communications, Tokyo, Japan
  • National Institute of Advanced Industrial Science and Technology, Tokyo, Japan
Pascal Lafourcade
  • Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Etienne, LIMOS, 63000 Clermont-Ferrand, France
Maxime Puys
  • Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Etienne, LIMOS, 63000 Clermont-Ferrand, France

Funding

This work was supported in part by JSPS KAKENHI Grant Numbers JP23H00479 and JP26K21164, by JSPS Bilateral Joint Research Projects JPJSBP120253206, by Institute of Mathematics for Industry, Joint Usage/Research Center in Kyushu University, 2023a020, 2024a035, and 2025a036, by PHC Sakura (53290WD), and by ANR Project PRIVA-SIQ (ANR-23-CE39-0008).

Acknowledgements

We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper.

References

  1. Yoshiki Abe, Mitsugu Iwamoto, and Kazuo Ohta. Efficient private PEZ protocols for symmetric functions. In Dennis Hofheinz and Alon Rosen, editors, Theory of Cryptography, volume 11891 of LNCS, pages 372-392, Cham, 2019. Springer. URL: https://doi.org/10.1007/978-3-030-36030-6_15.
  2. József Balogh, János A Csirik, Yuval Ishai, and Eyal Kushilevitz. Private computation using a PEZ dispenser. Theor. Comput. Sci., 306(1):69-84, 2003. URL: https://doi.org/10.1016/S0304-3975(03)00210-X.
  3. Xavier Bultel. Physical ring signature. In Andrei Z. Broder and Tami Tamir, editors, Fun with Algorithms, volume 291 of LIPIcs, pages 7:1-7:18. Schloss Dagstuhl, 2024. URL: https://doi.org/10.4230/LIPIcs.FUN.2024.7.
  4. Xavier Bultel, Jannik Dreier, Jean-Guillaume Dumas, and Pascal Lafourcade. Physical zero-knowledge proofs for Akari, Takuzu, Kakuro and KenKen. In Erik D. Demaine and Fabrizio Grandoni, editors, Fun with Algorithms, volume 49 of LIPIcs, pages 8:1-8:20, Dagstuhl, Germany, 2016. Schloss Dagstuhl. URL: https://doi.org/10.4230/LIPIcs.FUN.2016.8.
  5. Yu-Feng Chien and Wing-Kai Hon. Cryptographic and physical zero-knowledge proof: From Sudoku to Nonogram. In Paolo Boldi and Luisa Gargano, editors, Fun with Algorithms, volume 6099 of LNCS, pages 102-112, Berlin, Heidelberg, 2010. Springer. URL: https://doi.org/10.1007/978-3-642-13122-6_12.
  6. Jannik Dreier, Hugo Jonker, and Pascal Lafourcade. Secure auctions without cryptography. In Alfredo Ferro, Fabrizio Luccio, and Peter Widmayer, editors, Fun with Algorithms, volume 8496 of LNCS, pages 158-170. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-07890-8_14.
  7. Ronen Gradwohl, Moni Naor, Benny Pinkas, and Guy N. Rothblum. Cryptographic and physical zero-knowledge proof systems for solutions of Sudoku puzzles. Theory of Computing Systems, 44(2):245-268, 2009. URL: https://doi.org/10.1007/s00224-008-9119-9.
  8. Samuel Hand, Alexander Koch, Pascal Lafourcade, Daiki Miyahara, and Léo Robert. Efficient card-based ZKP for single loop condition and its application to Moon-or-Sun. New Gener. Comput., 42:449-477, 2024. URL: https://doi.org/10.1007/s00354-024-00274-1.
  9. Kyosuke Hatsugai, Suthee Ruangwises, Kyoichi Asano, and Yoshiki Abe. NP-completeness and physical zero-knowledge proofs for Sumplete, a puzzle generated by ChatGPT. New Gener. Comput., 42:429-448, 2024. URL: https://doi.org/10.1007/s00354-024-00267-0.
  10. Rie Ishikawa, Eikoh Chida, and Takaaki Mizuki. Efficient card-based protocols for generating a hidden random permutation without fixed points. In Cristian S. Calude and Michael J. Dinneen, editors, Unconventional Computation and Natural Computation, volume 9252 of LNCS, pages 215-226, Cham, 2015. Springer. URL: https://doi.org/10.1007/978-3-319-21819-9_16.
  11. Chuzo Iwamoto and Tatsuya Ide. Moon-or-Sun, Nagareru, and Nurimeizu are NP-complete. IEICE Trans. Fundamentals, 105(9):1187-1194, 2022. URL: https://doi.org/10.1587/transfun.2021DMP0006.
  12. Shohei Kaneko, Pascal Lafourcade, Lola-Baie Mallordy, Daiki Miyahara, Maxime Puys, and Kazuo Sakiyama. Balance-based ZKP protocols for pencil-and-paper puzzles. In Nicky Mouha and Nick Nikiforakis, editors, Information Security, volume 15257 of LNCS, pages 211-231, Cham, 2025. Springer. URL: https://doi.org/10.1007/978-3-031-75757-0_11.
  13. Shohei Kaneko, Pascal Lafourcade, Lola-Baie Mallordy, Daiki Miyahara, Maxime Puys, and Kazuo Sakiyama. Secure voting protocol using balance scale. In Kamel Adi, Simon Bourdeau, Christel Durand, Valérie Viet Triem Tong, Alina Dulipovici, Yvon Kermarrec, and Joaquín García-Alfaro, editors, Foundations and Practice of Security, volume 15532 of LNCS, pages 365-376, Cham, 2025. Springer. URL: https://doi.org/10.1007/978-3-031-87499-4_24.
  14. Alexander Koch and Stefan Walzer. Foundations for actively secure card-based cryptography. In Martin Farach-Colton, Giuseppe Prencipe, and Ryuhei Uehara, editors, Fun with Algorithms, volume 157 of LIPIcs, pages 17:1-17:23, Dagstuhl, Germany, 2020. Schloss Dagstuhl. URL: https://doi.org/10.4230/LIPIcs.FUN.2021.17.
  15. Yuichi Komano and Takaaki Mizuki. Coin-based secure computations. Int. J. Inf. Secur., 21:833-846, 2022. URL: https://doi.org/10.1007/s10207-022-00585-8.
  16. Yuichi Komano and Takaaki Mizuki. Physical zero-knowledge proof protocols for Topswops and Botdrops. New Gener. Comput., 42:399-428, 2024. URL: https://doi.org/10.1007/s00354-024-00272-3.
  17. Yuichi Komano and Takaaki Mizuki. Card-based zero-knowledge proof protocols for pancake sorting. IEICE Trans. Fundam., E109.A(3):371-382, 2026. URL: https://doi.org/10.1587/transfun.2025CIP0028.
  18. Pascal Lafourcade, Daiki Miyahara, Takaaki Mizuki, Léo Robert, Tatsuya Sasaki, and Hideaki Sone. How to construct physical zero-knowledge proofs for puzzles with a "single loop" condition. Theor. Comput. Sci., 888:41-55, 2021. URL: https://doi.org/10.1016/j.tcs.2021.07.019.
  19. Yuta Minamikawa and Kazumasa Shinagawa. Coin-based cryptographic protocols without hand operations. IEICE Trans. Fundamentals, E107.A(8):1178-1185, 2024. URL: https://doi.org/10.1587/transfun.2023EAP1082.
  20. Daiki Miyahara, Hiromichi Haneda, and Takaaki Mizuki. Card-based zero-knowledge proof protocols for graph problems and their computational model. In Qiong Huang and Yu Yu, editors, Provable and Practical Security, volume 13059 of LNCS, pages 136-152, Cham, 2021. Springer. URL: https://doi.org/10.1007/978-3-030-90402-9_8.
  21. Daiki Miyahara, Yuichi Komano, Takaaki Mizuki, and Hideaki Sone. Cooking cryptographers: Secure multiparty computation based on balls and bags. In Computer Security Foundations Symposium, pages 389-404, NY, 2021. IEEE. URL: https://doi.org/10.1109/CSF51468.2021.00034.
  22. Daiki Miyahara, Léo Robert, Pascal Lafourcade, and Shohei Kaneko. When locality implies globality: Card-based ZKP protocol for Shakashaka puzzle. In John Iacono, editor, Fun with Algorithms, volume 366 of LIPIcs, pages 22:1-22:15, Dagstuhl, Germany, 2026. Schloss Dagstuhl. URL: https://doi.org/10.4230/LIPIcs.FUN.2026.33.
  23. Daiki Miyahara, Léo Robert, Pascal Lafourcade, and Takaaki Mizuki. ZKP protocols for Usowan, Herugolf, and Five Cells. Tsinghua Science and Technology, 29(6):1651-1666, 2024. URL: https://doi.org/10.26599/TST.2023.9010153.
  24. Takaaki Mizuki and Hideaki Sone. Six-card secure AND and four-card secure XOR. In Xiaotie Deng, John E. Hopcroft, and Jinyun Xue, editors, Frontiers in Algorithmics, volume 5598 of LNCS, pages 358-369, Berlin, Heidelberg, 2009. Springer. URL: https://doi.org/10.1007/978-3-642-02270-8_36.
  25. Soma Murata, Daiki Miyahara, Takaaki Mizuki, and Hideaki Sone. Public-PEZ cryptography. In Willy Susilo, Robert H. Deng, Fuchun Guo, Yannan Li, and Rolly Intan, editors, Information Security, volume 12472 of LNCS, pages 59-74, Cham, 2020. Springer. URL: https://doi.org/10.1007/978-3-030-62974-8_4.
  26. Akihiro Nishimura, Yu-ichi Hayashi, Takaaki Mizuki, and Hideaki Sone. Pile-shifting scramble for card-based protocols. IEICE Trans. Fundam., 101(9):1494-1502, 2018. URL: https://doi.org/10.1587/transfun.E101.A.1494.
  27. Koji Nuida. Card-based protocol counting connected components of graphs. New Gener. Comput., 43:18, 2025. URL: https://doi.org/10.1007/s00354-025-00304-6.
  28. Taisei Otsuji, Peter Fulla, and Takuro Fukunaga. NP-Completeness and physical zero-knowledge proof of Hotaru Beam. In Yong Chen, Xiaofeng Gao, Xiaoming Sun, and An Zhang, editors, Computing and Combinatorics, pages 239-251, Singapore, 2024. Springer. URL: https://doi.org/10.1007/978-981-96-1090-7_20.
  29. Léo Robert, Daiki Miyahara, Pascal Lafourcade, and Takaaki Mizuki. Card-based ZKP for connectivity: Applications to Nurikabe, Hitori, and Heyawake. New Gener. Comput., 40:149-171, 2022. URL: https://doi.org/10.1007/s00354-022-00155-5.
  30. Léo Robert, Daiki Miyahara, Pascal Lafourcade, and Takaaki Mizuki. Physical ZKP protocols for Nurimisaki and Kurodoko. Theor. Comput. Sci., 972:114071, 2023. URL: https://doi.org/10.1016/j.tcs.2023.114071.
  31. Suthee Ruangwises. Physical zero-knowledge proof for ball sort puzzle. In Gianluca Della Vedova, Besik Dundua, Steffen Lempp, and Florin Manea, editors, Unity of Logic and Computation, volume 13967 of LNCS, pages 246-257, Cham, 2023. Springer. URL: https://doi.org/10.1007/978-3-031-36978-0_20.
  32. Suthee Ruangwises and Toshiya Itoh. Physical zero-knowledge proof for Ripple Effect. Theor. Comput. Sci., 895:115-123, 2021. URL: https://doi.org/10.1016/j.tcs.2021.09.034.
  33. Suthee Ruangwises and Toshiya Itoh. Physical ZKP for connected spanning subgraph: Applications to Bridges puzzle and other problems. In Irina Kostitsyna and Pekka Orponen, editors, Unconventional Computation and Natural Computation, pages 149-163, Cham, 2021,. Springer. URL: https://doi.org/10.1007/978-3-030-87993-8_10.
  34. Suthee Ruangwises and Toshiya Itoh. How to physically verify a rectangle in a grid: A physical ZKP for Shikaku. In Pierre Fraigniaud and Yushi Uno, editors, Fun with Algorithms, volume 226 of LIPIcs, pages 24:1-24:12, Dagstuhl, 2022. Schloss Dagstuhl. URL: https://doi.org/10.4230/LIPIcs.FUN.2022.24.
  35. Shun Sasaki and Kazumasa Shinagawa. Physical zero-knowledge proof for Sukoro. New Gener. Comput., 42:381-398, 2024. URL: https://doi.org/10.1007/s00354-024-00271-4.
  36. Tatsuya Sasaki, Daiki Miyahara, Takaaki Mizuki, and Hideaki Sone. Efficient card-based zero-knowledge proof for Sudoku. Theor. Comput. Sci., 839:135-142, 2020. URL: https://doi.org/10.1016/j.tcs.2020.05.036.
  37. Yuma Tamura, Akira Suzuki, and Takaaki Mizuki. Card-based zero-knowledge proof protocols for the 15-puzzle and the token swapping problem. In ACM ASIA Public-Key Cryptography Workshop, pages 11-22, New York, 2024. ACM. URL: https://doi.org/10.1145/3659467.3659905.
  38. Itaru Ueda, Daiki Miyahara, Akihiro Nishimura, Yu-ichi Hayashi, Takaaki Mizuki, and Hideaki Sone. Secure implementations of a random bisection cut. Int. J. Inf. Secur., 19(4):445-452, 2020. URL: https://doi.org/10.1007/s10207-019-00463-w.
  39. Ryuhei Uehara. Computational complexity of puzzles and related topics. Interdisciplinary Information Sciences, 29(2):119-140, 2023. URL: https://doi.org/10.4036/iis.2022.R.06.
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