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A geometric protocol for cryptography with cards

机译：卡加密的几何协议

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摘要

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.
机译：在广义的俄罗斯纸牌问题中，三名玩家Alice，Bob和Cath从a + b + c纸牌中分别抽出a，b和c卡。玩家只知道自己的纸牌以及纸牌组。然后，爱丽丝和鲍勃被要求通过公开消息相互交流他们的牌手。对于自然数k，如果Cath不了解爱丽丝是否持有给定的最多不是Cath的k张卡，则该通信被称为k安全。Swanson和Swanson最初将其称为弱k-安全性。斯汀森。 Atkinson提出的一种优雅解决方案将卡片视为有限投影平面上的点。尽管基于有限矢量空间而不是投影平面，但我们还是本着Atkinson精神提出了一种通用解决方案，并将其称为“几何协议”。给定任意c，k＞ 0，该协议为b = O（ac）的（a，b，c）的无限多值给出了广义的俄罗斯卡问题的有益且k安全的解决方案。这改善了已知解决方案的参数的收集。特别是，这是第一个在Cath拥有多个卡时保证k安全的解决方案。

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