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**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

Takuzu and Juosan are logical Nikoli games in the spirit of Sudoku. In Takuzu, a grid must be filled with 0’s and 1’s under specific constraints. In Juosan, the grid must be filled with vertical and horizontal dashes with specific constraints. We give physical algorithms using cards to realize zero-knowledge proofs for those games. The goal is to allow a player to show that he/she has the solution without revealing it. Previous work on Takuzu showed a protocol with multiple instances needed. We propose two improvements: only one instance needed and a soundness proof. We also propose a similar proof for Juosan game.

Daiki Miyahara, Léo Robert, Pascal Lafourcade, So Takeshige, Takaaki Mizuki, Kazumasa Shinagawa, Atsuki Nagao, and Hideaki Sone. Card-Based ZKP Protocols for Takuzu and Juosan. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{miyahara_et_al:LIPIcs.FUN.2021.20, author = {Miyahara, Daiki and Robert, L\'{e}o and Lafourcade, Pascal and Takeshige, So and Mizuki, Takaaki and Shinagawa, Kazumasa and Nagao, Atsuki and Sone, Hideaki}, title = {{Card-Based ZKP Protocols for Takuzu and Juosan}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {20:1--20:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.20}, URN = {urn:nbn:de:0030-drops-127817}, doi = {10.4230/LIPIcs.FUN.2021.20}, annote = {Keywords: Zero-knowledge proof, Card-based cryptography, Takuzu, Juosan} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

The satisfiability of a given branching program is to determine whether there exists a consistent path from the root to 1-sink.
In a syntactic read-k-times branching program, each variable appears at most k times in any path from the root to a sink.
We provide a satisfiability algorithm for syntactic read-k-times branching programs with n variables and m edges that runs in time O\left(\poly(n, m^{k^2})\cdot 2^{(1-\mu(k))n}\right), where \mu(k) = \frac{1}{4^{k+1}}. Our algorithm is based on the decomposition technique shown by Borodin, Razborov and Smolensky [Computational Complexity, 1993].

Atsuki Nagao, Kazuhisa Seto, and Junichi Teruyama. Satisfiability Algorithm for Syntactic Read-$k$-times Branching Programs. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 58:1-58:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{nagao_et_al:LIPIcs.ISAAC.2017.58, author = {Nagao, Atsuki and Seto, Kazuhisa and Teruyama, Junichi}, title = {{Satisfiability Algorithm for Syntactic Read-\$k\$-times Branching Programs}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {58:1--58:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.58}, URN = {urn:nbn:de:0030-drops-82423}, doi = {10.4230/LIPIcs.ISAAC.2017.58}, annote = {Keywords: branching program, read-k-times, satisfiability, moderately exponential time, polynomial space} }

Document

**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Toward the ultimate goal of separating L and P, Cook, McKenzie, Wehr, Braverman and Santhanam introduced the tree evaluation problem (TEP). For fixed h, k>0, FT_h(k) is given as a complete, rooted binary tree of height h, in which each internal node is associated with a function from [k]^2 to [k], and each leaf node with a number in [k]. The value of an internal node v is defined naturally, i.e., if it has a function f and the values of its two child nodes are a and b, then the value of v is f(a,b). Our task is to compute the value of the root node by sequentially executing this function evaluation in a bottom-up fashion. The problem is obviously in P and if we could prove that any branching program solving FT_h(k) needs at least k^(r(h)) states for any unbounded function r, then this problem is not in L, thus achieving our goal. The above authors introduced a restriction called thrifty against the structure of BP’s (i,e., against the algorithm for solving the problem) and proved that any thrifty BP needs Omega(k^h) states. This paper proves a similar lower bound for read-once branching programs, which allows us to get rid of the restriction on the order of nodes read by the BP that is the nature of the thrifty restriction.

Kazuo Iwama and Atsuki Nagao. Read-Once Branching Programs for Tree Evaluation Problems. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 409-420, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{iwama_et_al:LIPIcs.STACS.2014.409, author = {Iwama, Kazuo and Nagao, Atsuki}, title = {{Read-Once Branching Programs for Tree Evaluation Problems}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {409--420}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.409}, URN = {urn:nbn:de:0030-drops-44756}, doi = {10.4230/LIPIcs.STACS.2014.409}, annote = {Keywords: Lower bounds, Branching Programs, Read-Once Branching Programs, Space Complexity, Combinatorics} }

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