近年來，使用卡牌作為基礎，進行密碼學研究的領域逐漸擴大。這種方法仰賴於一疊實體卡牌來執行各種加密功能，涵蓋了從基本的安全計算(例如安全 的 AND, OR 和 XOR 等布林運算)到更複雜的任務(例如安全排名、零知識證 明、安全排序和安全投票)等等。
在本論文中，我們對前人研究的安全投票進行了一般化，目的是找出唯一的優勝候選人。在同樣適用於任意數量的投票者中，我們將投票系統從僅有兩 個候選人的情況，擴展成可以容許任意數量的候選人。根據投票數量的限制，我們制定了兩種情況及其對應的演算法。在過程中，我們也利用了姚氏百萬富翁問題的解法，創造出一種方法讓任意的基於比較的排序可以使用卡牌協議進行。此外，我們依據制定的安全投票規則，建立了平行化的模型，並對兩種方法進行了平行操作，以改進演算法的執行時間。最後，我們引入了緩衝區的概念，將卡牌區分為唯讀卡牌和工作卡牌，提出了一個新的概念，將來或許能夠幫助分析其他卡牌協議的效能。

Recently, the research of card-based cryptography, which relies on a deck of physical cards to perform cryptographic functionalities, has been growing. It spans from basic secure computations, such as secure AND, OR, and XOR Boolean functions, to more complex tasks, such as secure ranking, zero-knowledge proof, secure sorting, and secure voting.
In this thesis, we generalize the concept of secure voting from previous research, with the aim of determining the unique winning candidate. We extend the voting system from a scenario that only handles two candidates, to one that can accommodate any number of candidates for an arbitrary number of voters. Depending on the number of ballots, we formulate two scenarios and design corresponding algorithms. In the process of designing the methods, we also leverage the solution from Yao’s Millionaire Problem to create a method that allows any comparison-based sorting to be performed using card-based protocols. Additionally, we establish a parallelized model based on the defined secure voting rules and perform parallel operations on the two methods to improve the algorithm’s execution time. Finally, we introduce the concept of working memory, dividing the cards into read-only cards and working cards, proposing a new concept that may aid in analyzing the efficiency of other card-based protocols.

[1] Yoshiki Abe, Takeshi Nakai, Yoshihisa Kuroki, Shinnosuke Suzuki, Yuta Koga, Yohei Watanabe, Mitsugu Iwamoto, and Kazuo Ohta. Efficient Card- Based Majority Voting Protocols. New Generation Computing, 40(1):173–198, 2022.
[2] Bert den Boer. More Efficient Match-Making and Satisfiability: The Five Card Trick. In Proceedings of Advances in Cryptology (EUROCRYPT), pages 208–217, 1989.
[3] Rikuo Haga, Kodai Toyoda, Yuto Shinoda, Daiki Miyahara, Kazumasa Shi- nagawa, Yuichi Hayashi, and Takaaki Mizuki. Card-Based Secure Sorting Protocol. In Proceedings of Advances in Information and Computer Security, pages 224–240, 2022.
[4] Daiki Miyahara, Yu-ichi Hayashi, Takaaki Mizuki, and Hideaki Sone. Prac- tical Card-Based Implementations of Yao’s Millionaire Protocol. Theoretical Computer Science (TCS), 803(11):207–221, 2019.
[5] Daiki Miyahara, Itaru Ueda, Yu-ichi Hayashi, Takaaki Mizuki, and Hideaki Sone. Analyzing Execution Time of Card-Based Protocols. In Proceedings of Unconventional Computation and Natural Computation (UCNC), pages 145– 158, 2018.
26
[6] Takaaki Mizuki, Isaac Kobina Asiedu, and Hideaki Sone. Voting with a Log- arithmic Number of Cards. In Proceedings of Unconventional Computation and Natural Computation (UCNC), pages 162–173, 2013.
[7] Takaaki Mizuki and Hideaki Sone. Six-Card Secure AND and Four-Card Secure XOR. In Proceedings of Frontiers in Algorithmics (FAW), pages 358– 369, 2009.
[8] Takuya Nishida, Takaaki Mizuki, and Hideaki Sone. Securely Computing the Three-Input Majority Function with Eight Cards. In Proceedings of Theory and Practice of Natural Computing, pages 193–204, 2013.
[9] Tatsuya Sasaki, Daiki Miyahara, Takaaki Mizuki, and Hideaki Sone. Efficient Card-Based Zero-Knowledge Proof for Sudoku. Theoretical Computer Science (TCS), 839(5):135–142, 2020.
[10] Kazumasa Shinagawa and Takaaki Mizuki. Secure Computation of Any Boolean Function based on Any Deck of Cards. In Proceedings of Frontiers in Algorithmics (FAW), pages 63–75, 2019.
[11] Ken Takashima, Yuta Abe, Tatsuya Sasaki, Daiki Miyahara, Kazumasa Shi- nagawa, Takaaki Mizuki, and Hideaki Sone. Card-Based Secure Ranking Computations. In Proceedings of Combinatorial Optimization and Applica- tions (COCOA), pages 461–472, 2019.
[12] Itaru Ueda, Akihiro Nishimura, Yu-ichi Hayashi, Takaaki Mizuki, and Hideaki Sone. How to Implement a Random Bisection Cut. In Proceedings of Theory and Practice of Natural Computing, pages 58–69, 2016.