Necessary and Sufficient Numbers of Cards for the Transformation Protocol

Koizumi, Koichi; Mizuki, Takaaki; Nishizeki, Takao

doi:10.1007/978-3-540-27798-9_12

Cited by 5 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some are interested in card deal protocols that allow players to agree on a common secret without a given eavesdropper being able to determine this secret value. This area of research is especially interesting in terms of possible applications to key generation; see, for example [10,11,12,13,14,17,16,2]. Others are concerned with analyzing variations of the problem using epistemic logic [18,19,20,7].…”

mentioning

confidence: 99%

Combinatorial solutions providing improved security for the generalized Russian cards problem

Swanson

Stinson

2012

Des. Codes Cryptogr.

View full text Add to dashboard Cite

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. * D. Stinson's research is supported by NSERC discovery grant • We distinguish between deterministic strategies, in which the hand H A held by Alice uniquely determines the index i that she will broadcast, and non-deterministic, possibly even biased announcement strategies. We are especially interested in strategies with uniform probability distributions, which we will refer to as equitable.n−a+c c different announcements. However, there are a total of n−a+c c announcements, so every announcement must contain a block that contains X ′ .An optimal (3, 3, 1)-strategy would have m = 5. From Theorem 2.4, the existence of such a strategy would be equivalent to a large set of five STS(7). As mentioned above, it is known that this large set does not exist. However, from Example 1.1, we obtain a (3, 3, 1)-strategy for Alice with m = 6 that is informative for Bob. Thus we have proven the following.Theorem 2.5. The minimum m such that there exists a (3, 3, 1)-strategy for Alice that is informative for Bob is m = 6.

show abstract

mentioning

confidence: 99%

Combinatorial solutions providing improved security for the generalized Russian cards problem

Swanson

Stinson

2012

Des. Codes Cryptogr.

View full text Add to dashboard Cite

show abstract

“…The Russian cards problem and variants of it has received a fair amount of attention in the literature, with focus ranging from possible applications to key generation [2,3,[15][16][17][18][19]21,23], to analyses based on epistemic logic [9][10][11][12], to card deals with more than three players [14,20]. Of more relevance to our work is the recent research that takes a combinatorial approach [1-4, 6, 27], on which we now focus.…”

Section: Discussion and Comparison With Related Workmentioning

confidence: 99%

Additional Constructions to Solve the Generalized Russian Cards Problem using Combinatorial Designs

Swanson

Stinson

2014

Electron. J. Combin.

View full text Add to dashboard Cite

In the generalized Russian cards problem, we have a card deck X of n cards and three participants, Alice, Bob, and Cathy, dealt a, b, and c 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, for her part, should remain ignorant of all * Much of this work appears in the PhD thesis of the first author [28]. 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 1-security) to not gaining any probabilistic advantage in guessing the fate of some set of δ cards (perfect δ-security). As we demonstrate in this work, 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 δ-secure strategies and (c + δ)-designs on n points with block size a, 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 1-security against Cathy when c = 2. Drawing on our equivalence results, we are able to use a known combinatorial design to construct a strategy with a = 8, b = 13, and c = 3 that is perfectly 2-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 δ-security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of c = ⌊ a−δ 2 ⌋ cards. Alternatively, the construction yields solutions for arbitrary δ, c and any a ≥ δ + 2c.

show abstract

“…The solution to the problem will imply a method to communicate information among parties in a distributed computing setting securely without using any encryption [15], [23], [20]. The analogy is that, the communicating agents and adversaries are modeled as players and the information to be communicated as the ownership of cards.…”

Section: Introductionmentioning

confidence: 99%

“…The analogy is that, the communicating agents and adversaries are modeled as players and the information to be communicated as the ownership of cards. It is generally believed that the above game gives unconditional security [11], [15], [23], [20].…”

Section: Introductionmentioning

confidence: 99%

Lower Bound for the Communication Complexity of the Russian Cards Problem

Cyriac,

Krishnan

2008

Preprint

View full text Add to dashboard Cite

In this paper it is shown that no public announcement scheme that can be modeled in Dynamic Epistemic Logic (DEL) can solve the Russian Cards Problem (RCP) in one announcement. Since DEL is a general model for any public announcement scheme [11], [3], [6], [21], [12] we conclude that there exist no single announcement solution to the RCP. The proof demonstrates the utility of DEL in proving lower bounds for communication protocols. It is also shown that a general version of RCP has no two announcement solution when the adversary has sufficiently large number of cards.

show abstract

Necessary and Sufficient Numbers of Cards for the Transformation Protocol

Cited by 5 publications

References 7 publications

Combinatorial solutions providing improved security for the generalized Russian cards problem

Combinatorial solutions providing improved security for the generalized Russian cards problem

Additional Constructions to Solve the Generalized Russian Cards Problem using Combinatorial Designs

Lower Bound for the Communication Complexity of the Russian Cards Problem

Contact Info

Product

Resources

About