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Combinatorial solutions providing improved security for the generalized Russian cards problem

机译：组合解决方案为广义的俄罗斯卡问题提供了改进的安全性

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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 ≥ 2, where b≥ 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 ＞ d - 1 involving d - (n,d + 1, 1 )-designs.
机译：我们提出了广义俄罗斯卡问题的第一个正式数学表示，并提供了严格的安全性定义，可以捕获弱和完美安全性概念的基本版本和扩展版本。在普遍的俄罗斯纸牌问题中，向三名玩家Alice，Bob和Cathy发了n张纸牌，分别给了a，b和c卡。我们的目标是让爱丽丝和鲍勃通过公共交流互相学习对方的手，而不让凯茜了解任何特定卡片的命运。基本思想是，爱丽丝宣布自己可能握住的一组手牌，而鲍勃利用自己的手知识，应该能够从该公告中学习爱丽丝的牌，但凯茜则不能。使用组合方法，我们能够很好地描述信息策略（即使Bob学会Alice的手的策略），并具有最佳的通信复杂性，即Alice宣布的可能手的集合必须等于t的大集合。 -（n，a，1）个设计，其中t = a-c。我们还为某些类型的交易同时提供信息和安全性提供了一些有趣的必要条件。也就是说，对于满足d≥2的c = a-d的交易（其中b≥d-1）并且假定该策略满足强大的安全性版本（即完美（d-1）-安全性），我们证明a = d + 1，因此c =1。我们还精确地描述了满足b＞ d-1且涉及d-的形式（d + 1，b，1）的信息安全交易（d-1）的完美特征。 （n，d + 1，1）个设计。

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