Abstract
In the generalized Russian cards problem, the three players Alice, Bob and Cath draw \(a,b\) and \(c\) cards, respectively, from a deck of \(a+b+c\) cards. Players only know their own cards and what the deck of cards is. Alice and Bob are then required to communicate their hand of cards to each other by way of public messages. For a natural number \(k\), the communication is said to be \(k\)-safe if Cath does not learn whether or not Alice holds any given set of at most \(k\) cards that are not Cath’s, a notion originally introduced as weak \(k\)-security by Swanson and Stinson. An elegant solution by Atkinson views the cards as points in a finite projective plane. We propose a general solution in the spirit of Atkinson’s, although based on finite vector spaces rather than projective planes, and call it the ‘geometric protocol’. Given arbitrary \(c,k>0\), this protocol gives an informative and \(k\)-safe solution to the generalized Russian cards problem for infinitely many values of \((a,b,c)\) with \(b=O(ac)\). This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees \(k\)-safety when Cath has more than one card.
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Notes
Note however that this restriction may be circumvented by using protocols of more than two steps [4].
Our presentation follows that given in [17]. Compare to [1, 18], where informative for \((A,B,C)\) is defined as follows: Given a deal \((A,B,C)\), an announcement \(\mathcal A \) containing \(A\) is informative for \((A,B,C)\) if for any \(A^{\prime } \in \mathcal A \) and any \((A^{\prime },B^{\prime },C^{\prime })\), there is only one \(X\in \mathcal A \) such that \(X\subseteq A^{\prime }\cup C^{\prime }\). In principle this is stronger than the definition we give; however, this is remedied by the more general condition of being informative for \((a,b,c)\).
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Acknowledgments
We acknowledge support from the project ERC Starting Grant EPS 313360, the project FFI2011-15945-E (Ministerio de Economía y Competitividad, Spain) and the Excellence Research Project of the Junta de Andalucía P08-HUM-04159. Hans van Ditmarsch is also affiliated to IMSc, Chennai, India, as research associate. We would also like to express our gratitude to the anonymous referees.
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Cordón-Franco, A., van Ditmarsch, H., Fernández-Duque, D. et al. A geometric protocol for cryptography with cards. Des. Codes Cryptogr. 74, 113–125 (2015). https://doi.org/10.1007/s10623-013-9855-y
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DOI: https://doi.org/10.1007/s10623-013-9855-y