A geometric protocol for cryptography with cards

Abstract

In the generalized Russian cards problem, the three players Alice, Bob and Cath draw a, b and c cards, respectively, from a deck of 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. The communication is said to be safe if Cath does not learn the ownership of any specific card; in this paper we consider a strengthened notion of safety introduced by Swanson and Stinson which we call k-safety.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 c, k > 0, this protocol gives an informative and k-safe solution to the generalized Russian cards problem for infinitely many values of (a, b, c) with b = O(ac). This improves on the collection of parameters for which solutions are known. In particular, it is the first solution which guarantees k-safety when Cath has more than one card.

Perfectly secure data aggregation via shifted projections

“…The latter may be traced back to [7] but has recently received renewed attention [14], leading to many new solutions (e.g. [1,3,13]). In the original Russian cards problem, there are only two communicating agents.…”

“…Compare this to the protocol presented in[3], where it is Alice's hand that would form a line rather than its complement.…”

Secure aggregation of distributed information: How a team of agents can safely share secrets in front of a spy

“…We will show the SADI problem for distribution type (3, 3, 1) is solvable, by informally describing the following solving protocol. It is a variation of a solution to the two-agent Russian cards problem, which appeared in [2] and, in a presentation closer to ours, in [4]. Alice 'places' all cards in the points of the 7-point projective plane, also known as the Fano plane, in such a way that her cards form a line, as indicated in Figure 1.…”

A case study in almost-perfect security for unconditionally secure communication

“…This strategy generalizes that in [2] where α = δ − 1, although that article also considers the case where Alice holds more than one plane. Let us now show that the strategy is equitable.…”