Three Steps

Given is a deal of ten cards over three players, such that two players each get four cards and the remaining player (the ‘eavesdropper’) two cards. We show that there does not exist a protocol of two steps for the four-card players to inform each other safely of their hands of cards, and we then present a protocol of three steps that achieves that goal. We verify the properties of that protocol by combinatorial and, mainly, logical (model checking) means. No such three-step protocol for cards was known. The method can be generalized. This will advance the characterization of card deals for which such exchanges of secrets are possible. 1 Knowledge-based protocols for card players From a pack of seven known cards two players A and B each draw three cards and a third player C gets the remaining card. How can A and B openly (publicly) inform each other about their cards, without the third player learning from any of their cards who holds it? This problem is often known as the Russian Cards Problem [16] and goes back to [9]. Such issues in cards cryptography have been investigated in the logical and model checking community [19, 4, 17, 11], including related issues in epistemic puzzles, and also in combinatorics and theoretical computer science [7, 14, 10, 20], including related issues in bit-exchange protocols. One solution for the riddle is as follows. Suppose that the actual deal of cards is that agent A has {0, 1, 2}, B has {3, 4, 5} and C has {6}. – A says: My hand is one of 012, 046, 136, 145, 235. – B says: C’s card is 6. After this, it is common knowledge to the three agents that A knows the hand of B, that B knows the hand of A, and that C is ignorant of the ownership of any card not held by herself. We can also see these two sequences as the execution of a knowledge-based protocol. Given A’s hand of cards, there is a (non-deterministic) way to produce her announcement, and given her announcement, B always responds by announcing C’s card. The protocol is knowledge-based, because the agents initially only know their own hand of cards, and have public knowledge of the deck of cards and how many cards each agent has drawn from the pack. We can imagine agent A producing her announcement as follows from her initial knowledge (so, indeed, it is a function of her local state to her action):