### Combinatorial Solutions Providing Improved Security for the Generalized Russian Cards Problem

Colleen M. Swanson and Douglas R. Stinson

##### Abstract

We present the first formal mathematical presentation of the generalized Russian cards problem, and provide rigorous security definitions that capture both basic and extended versions of weak and perfect security notions. In the generalized Russian cards problem, three players, Alice, Bob, and Cathy, are dealt a deck of $n$ cards, each given $a$, $b$, and $c$ cards, respectively. The goal is for Alice and Bob to learn each other's hands via public communication, without Cathy learning the fate of any particular card. The basic idea is that Alice announces a set of possible hands she might hold, and Bob, using knowledge of his own hand, should be able to learn Alice's cards from this announcement, but Cathy should not. Using a combinatorial approach, we are able to give a nice characterization of informative strategies (i.e., strategies allowing Bob to learn Alice's hand), having optimal communication complexity, namely the set of possible hands Alice announces must be equivalent to a large set of $t-(n, a, 1)$-designs, where $t=a-c$. We also provide some interesting necessary conditions for certain types of deals to be simultaneously informative and secure. That is, for deals satisfying $c = a-d$ for some $d \geq 2$, where $b \geq d-1$ and the strategy is assumed to satisfy a strong version of security (namely perfect $(d-1)$-security), we show that $a = d+1$ and hence $c=1$. We also give a precise characterization of informative and perfectly $(d-1)$-secure deals of the form $(d+1, b, 1)$ satisfying $b \geq d-1$ involving $d-(n, d+1, 1)$-designs.

Available format(s)
Category
Cryptographic protocols
Publication info
Published elsewhere. Unknown where it was published
Keywords
information theoryRussian Cards problemcombinatorial designs
Contact author(s)
c2swanso @ uwaterloo ca
History
Short URL
https://ia.cr/2012/376 CC BY

BibTeX

@misc{cryptoeprint:2012/376,
author = {Colleen M.  Swanson and Douglas R.  Stinson},
title = {Combinatorial Solutions Providing Improved Security for the Generalized Russian Cards Problem},
howpublished = {Cryptology ePrint Archive, Paper 2012/376},
year = {2012},
note = {\url{https://eprint.iacr.org/2012/376}},
url = {https://eprint.iacr.org/2012/376}
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