Dealing Necessary and Sufficient Numbers of Cards for Sharing a One-Bit Secret Key (Extended Abstract)

Abstract

Abstract. Using a random deal of cards to players and a computationally unlimited eavesdropper, all players wish to share a one-bit secret key which is information-theoretically secure from the eavesdropper. This can be done by a protocol to make several pairs of players share one-bit secret keys so that all these pairs form a spanning tree over players. In this paper we obtain a necessary and sufficient condition on the number of cards for the existence of such a protocol. Our condition immediately yields an efficient linear-time algorithm to determine whether there exists a protocol to achieve such a secret key sharing.

Untitled

“…Since δ = 1, by Eq. (10) we have h = c e + k − f = c e + (k − 1) − (k − 2) = c e + 1 = h. Therefore by Eq. (6) we have = max{min{c 2 − h , b }, 0} = max{min{c 2 − h, b − 1}, 0} = .…”

Necessary and Sufficient Numbers of Cards for Sharing Secret Keys on Hierarchical Groups

“…Let $W$ be the set of all signatures for each of which there is akey set protocol working, and let $L$ be the set of all signatures for each of which there is no key set protocol working. Asimple necessary and sufficient condition for $\gamma$ $\in W$ has been known $ [2,8]$ . Furthermore, acharacterization of "optimal" key set protocols is given in [7].…”

“…They also give asufficient condition on the numbers of cards for the "key set protocol" to always form atree. Mizuki et al give a simple necessary and sufficient condition on the numbers of cards for the "key set protocol" to always form atree [8].…”

Characterization of Optimal Key Set Protocols

“…The set C and the signature γ are public to all the players and even to Eve, but the cards in the hand of a player or Eve are private to herself, as in the case of usual card games. This paper addresses protocols which make all the players share a common one-bit secret key information-theoretically securely using such a random deal of cards [2,3,4,5,6,10]. A reasonable situation in which such protocols are practically required is discussed in [4,6], and also the reason why we deal cards even to Eve is found there.…”

“…γ ∈ Γ 2 , Fischer and Wright give a simple necessary and sufficient condition for γ ∈ W [3]. For k ≥ 3, the authors give a simple necessary and sufficient condition for γ ∈ W [10]. (These necessary and sufficient conditions will be described in Section 2.5.…”