A geometric protocol for cryptography with cards
This addresses secure communication in card-based cryptography for scenarios with multiple eavesdroppers, offering an incremental improvement over existing methods.
The paper tackles the generalized Russian cards problem by proposing a geometric protocol based on finite vector spaces, which provides an informative and k-safe solution for infinitely many parameter sets, improving on known solutions and being the first to guarantee k-safety when Cath has more than one card.
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.