A geometric protocol for cryptography with cards

TL;DR

This paper introduces a geometric protocol based on finite vector spaces that provides safe and informative communication solutions for the generalized Russian cards problem, especially when Cath holds multiple cards.

Contribution

It presents a novel geometric protocol using finite vector spaces that extends safety guarantees to cases where Cath has multiple cards, improving known parameter ranges.

Findings

Provides a solution for infinitely many (a,b,c) with b=O(ac)

Guarantees k-safety for Cath with multiple cards

Extends the known parameters for safe communication protocols

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.