Abstract
The two sheriffs problem is the following problem. There are two sheriffs, and each of them has their own list of suspects. Assuming that these lists are the result of a proper investigation, we can say that a culprit is the intersection of them even if the sheriffs do not know who the culprit is. Now, they wish to identify the culprit through an open channel, i.e., to compute the intersection of two lists, without letting an eavesdropper know the culprit who observed all communications. This cryptographic problem was proposed by Beaver et al., and a combinatorial solution using a bipartite graph was proposed. In this paper, we propose a formulation of the two sheriffs problem by introducing a secrecy evaluation based on the eavesdropper’s attack success probability. Furthermore, we propose an improved version of Beaver et al.’s protocol that an arbitrary number of players can execute and has less attack success probability.
This work was supported by JSPS KAKENHI Grant Numbers JP23H00468, JP23H00479, JP23K17455, JP23K16880, JP22H03590, JP21H03395, JP21H03441, JP18H05289, and MEXT Leading Initiative for Excellent Young Researchers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
To be precise, the attack success probability of the third protocol is at most \((|\mathcal {A}|-1)/|\mathcal {A}|\), and it means that it depends on the size of \(|\mathcal {A}|\) (also see Table 1). Nonetheless, it holds \((|\mathcal {A}|-1)/|\mathcal {A}| \ge 1/2\) for any \(\mathcal {A}\) s.t. \(|\mathcal {A}|\ge 2\).
- 2.
Even if it holds that \(\mathcal {S}_k \not \subset \widetilde{\mathcal {S}}\) for some \(k \in \{2,\ldots ,n\}\), it does not matter; the protocol works well since \(\widetilde{\mathcal {S}}\) includes \(\mathcal {S}_1\), which also includes \(\bigcap _{k=1}^{n}{\mathcal {S}_k}\).
References
Beaver, D., Haber, S., Winkler, P.: On the isolation of a common secret. Math. Paul Erdös II 121–135 (1997)
Brody, J., Jakobsen, S.K., Scheder, D., Winkler, P.: Cryptogenography. In: ITCS, pp. 13–22 (2014)
Doerr, B., Kunnemann, M.: Improved protocols and hardness results for the two-player cryptogenography problem. IEEE Trans. Inf. Theory 66(9), 5729–5741 (2020). https://doi.org/10.1109/tit.2020.2978385. https://ieeexplore.ieee.org/document/9115678/
Jakobsen, S.K.: Information theoretical cryptogenography. J. Cryptol. 30(4), 1067–1115 (2016). https://doi.org/10.1007/s00145-016-9242-8.pdf
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Sugimoto, K., Nakai, T., Watanabe, Y., Iwamoto, M. (2024). The Two Sheriffs Problem: Cryptographic Formalization and Generalization. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14461. Springer, Cham. https://doi.org/10.1007/978-3-031-49611-0_37
Download citation
DOI: https://doi.org/10.1007/978-3-031-49611-0_37
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-49610-3
Online ISBN: 978-3-031-49611-0
eBook Packages: Computer ScienceComputer Science (R0)