Abstract
Alice and Bob wish to privately exchange information by public announcements overheard by Cath. To do so by a deterministic protocol, their inputs must be correlated. Dependent inputs are represented using a deck of cards. There is a publicly known signature \((\mathbf {a},\mathbf {b},\mathbf {c})\), meaning that A gets \(\mathbf {a}\) cards, B gets \(\mathbf {b}\) cards, and C gets \(\mathbf {c}\) cards, out of the deck of \({n}\) cards, \({n}=\mathbf {a}+\mathbf {b}+\mathbf {c}+\mathbf {r}\). We use a perspective inspired by distributed computing that considers colorings of a generalization of Johnson graphs, together with techniques based on Singer difference sets and shifting, to study the classic Russian cards problem \(\mathbf {a}=\mathbf {b}=3\), \(\mathbf {c}=1\), and \(\mathbf {r}=0\). We consider also a novel variant where they wish to learn something about each other’s hands. We focus on the number of bits that Alice and Bob need to exchange to solve either the classic or the minimally informative version of the problem.
Supported by the UNAM-PAPIIT project IN106520.
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Notes
- 1.
First A sends the sum of her cards modulo 7. Using this information, B can deduce the hand of A, and thus responds by announcing the card of C.
- 2.
Notice that when C listens to a message M, say from A, she knows that the hand of A could be any a such that \(P_A(a)=M\). Thus, M can be seen as an encoding of all such hands a, as in some previous papers e.g. [4].
- 3.
The set of maximal cliques in J(n, m) have been well-studied, they are of size \(n-m+1\) and \(m+1\) e.g. [13].
- 4.
We also denote the singleton set with card c as c, as it is always clear from the context which case it is.
- 5.
The same lower bound is [22, Theorem 4], proved by reduction to a combinatorial design theorem.
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We thank Hans van Ditmarsch for his comments.
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Leyva-Acosta, Z., Pascual-Aseff, E., Rajsbaum, S. (2021). Information Exchange in the Russian Cards Problem. In: Johnen, C., Schiller, E.M., Schmid, S. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2021. Lecture Notes in Computer Science(), vol 13046. Springer, Cham. https://doi.org/10.1007/978-3-030-91081-5_25
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