In the generalized Russian cards problem, we have a card deck @math of @math cards and three participants, Alice, Bob, and Cathy, dealt @math , @math , and @math cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (weak @math -security) to not gaining any probabilistic advantage in guessing the fate of some set of @math cards (perfect @math -security). As we demonstrate, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly @math -secure strategies and @math -designs on @math points with block size @math , when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect @math -security against Cathy when @math . We leverage a known combinatorial design to construct a strategy with @math , @math , and @math that is perfectly @math -secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak @math -security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of @math cards. Alternatively, the construction yields solutions for arbitrary @math , @math and any @math .
Cord ' o n- @cite_6 consider four-step solutions that achieve weak @math -security for the generalized Russian cards problem with parameters @math such that @math ; this is the first work that shows it is possible to achieve weak @math -security in cases where Cathy holds more cards than one of the other players. The authors demonstrate the existence of a necessary construction for Bob's announcement when the card deal parameters satisfy specific conditions and briefly address the feasibility of finding such constructions in practice. In particular, the authors leave as an interesting open problem efficient algorithms for producing Bob's announcement.
@cite_6
1596450573 2141368734 2010404638 2073060670
In the generalized Russian cards problem, we have a card deck @math of @math cards and three participants, Alice, Bob, and Cathy, dealt @math , @math , and @math cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (weak @math -security) to not gaining any probabilistic advantage in guessing the fate of some set of @math cards (perfect @math -security). As we demonstrate, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly @math -secure strategies and @math -designs on @math points with block size @math , when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect @math -security against Cathy when @math . We leverage a known combinatorial design to construct a strategy with @math , @math , and @math that is perfectly @math -secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak @math -security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of @math cards. Alternatively, the construction yields solutions for arbitrary @math , @math and any @math . 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 @math cards, each given @math , @math , and @math 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 @math -designs, where @math . We also provide some interesting necessary conditions for certain types of deals to be simultaneously informative and secure. That is, for deals satisfying @math for some @math , where @math and the strategy is assumed to satisfy a strong version of security (namely perfect @math -security), we show that @math and hence @math . We also give a precise characterization of informative and perfectly @math -secure deals of the form @math satisfying @math involving @math -designs. In the generalized Russian cards problem, the three players Alice, Bob and Cath draw @math a , b and @math c cards, respectively, from a deck of @math 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 @math k , the communication is said to be @math k -safe if Cath does not learn whether or not Alice holds any given set of at most @math k cards that are not Cath's, a notion originally introduced as weak @math 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 @math c , k > 0 , this protocol gives an informative and @math k -safe solution to the generalized Russian cards problem for infinitely many values of @math ( a , b , c ) with @math b = O ( a c ) . This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees @math k -safety when Cath has more than one card. In the generalized Russian cards problem, Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of size a+b+c. Alice and Bob must then communicate their entire hand to each other, without Cath learning the owner of a single card she does not hold. Unlike many traditional problems in cryptography, however, they are not allowed to encode or hide the messages they exchange from Cath. The problem is then to find methods through which they can achieve this. We propose a general four-step solution based on finite vector spaces, and call it the ''colouring protocol'', as it involves colourings of lines. Our main results show that the colouring protocol may be used to solve the generalized Russian cards problem in cases where a is a power of a prime, c=O(a^2) and b=O(c^2). This improves substantially on the set of parameters for which solutions are known to exist; in particular, it had not been shown previously that the problem could be solved in cases where the eavesdropper has more cards than one of the communicating players.