Abstract
We consider a problem, which we call secure grouping, of dividing a number of parties into some subsets (groups) in the following manner: Each party has to know the other members of his/her group, while he/she may not know anything about how the remaining parties are divided (except for certain public predetermined constraints, such as the number of parties in each group). In this paper, we construct an information-theoretically secure protocol using a deck of physical cards to solve the problem, which is jointly executable by the parties themselves without a trusted third party. Despite the non-triviality and the potential usefulness of the secure grouping, our proposed protocol is fairly simple to describe and execute. Our protocol is based on algebraic properties of conjugate permutations. A key ingredient of our protocol is our new techniques to apply multiplication and inverse operations to hidden permutations (i.e., those encoded by using face-down cards), which would be of independent interest and would have various potential applications.
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Notes
- 1.
In some card games, the dealer announces one of the cards (e.g., “
”) and then the player having this card becomes the dealer’s partner. However, now the dealer cannot know who is the partner, though the partner him/herself can know that he/she is the dealer’s partner; hence the condition of secure grouping is not achieved. - 2.
Note that this sequence of number cards is obtained by moving, for each \(k = 1,2,\dots ,7\), the k-th card
to the \(\sigma (k)\)-th position. For example, if \(\sigma (k) = k+1\) for \(1 \le k \le 6\) and \(\sigma (7) = 1\), then the resulting card sequence is 
. - 3.
Usually, we define coding rules such as
and 
since the card-based protocol normally uses Boolean values. If the usual Boolean encoding rule is used instead of the number cards, the secure grouping protocol can still be executed. In the case the number of cards increases \(2\lceil \log _2{n} \rceil \) times larger.
”) and then the player having this card becomes the dealer’s partner. However, now the dealer cannot know who is the partner, though the partner him/herself can know that he/she is the dealer’s partner; hence the condition of secure grouping is not achieved.
to the 
.
and 
since the card-based protocol normally uses Boolean values. If the usual Boolean encoding rule is used instead of the number cards, the secure grouping protocol can still be executed. In the case the number of cards increases References
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Acknowledgement
We thank the members of Shin-Akarui-Angou-Benkyou-Kai for their helpful comments. In particular we would like to thank Shuichi Katsumata for his helpful comments. A part of this work is supported by JST CREST grant number JPMJCR1688.
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Hashimoto, Y., Shinagawa, K., Nuida, K., Inamura, M., Hanaoka, G. (2017). Secure Grouping Protocol Using a Deck of Cards. In: Shikata, J. (eds) Information Theoretic Security. ICITS 2017. Lecture Notes in Computer Science(), vol 10681. Springer, Cham. https://doi.org/10.1007/978-3-319-72089-0_8
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