Abstract
Norinori is a logic game similar to Sudoku. In Norinori, a grid of cells has to be filled with either black or white cells so that the given areas contain exactly two black cells, and every black cell shares an edge with exactly one other black cell. We propose a secure interactive physical algorithm, relying only on cards, to realize a zero-knowledge proof of knowledge for Norinori. It allows a player to show that he or she knows a solution without revealing it. For this, we show in particular that it is possible to physically prove that a particular element is present in a list, without revealing any other value in the list, and without revealing the actual position of that element in the list.
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Notes
- 1.
It means that the cards face down are indistinguishable from each other.
- 2.
For example, if the size of the puzzle board is \(3\times 4\), the north card is the fifth card away from the chosen card to the left, and the south card is the sixth card away from it to the right.
- 3.
One might think that t times would suffice if any black card found in Step 5 was replaced by a marker card; however, this is not the case because we need to check that such a found black card also has exactly one black card among its adjacent cards.
- 4.
This trick is inspired from [14] and allows us to also have no soundness error.
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Acknowledgement
We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper. This work was supported by JSPS KAKENHI Grant Number JP17K00001 and partly by the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541) and the Cyber@Alps French National Research Agency program (ANR-15-IDEX-02).
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Dumas, JG., Lafourcade, P., Miyahara, D., Mizuki, T., Sasaki, T., Sone, H. (2019). Interactive Physical Zero-Knowledge Proof for Norinori. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_14
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