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

TL;DR

This paper introduces an intermediate security notion called ε-strong security for the Russian cards problem, demonstrating that a modified geometric strategy can achieve near-perfect security with arbitrarily small ε.

Contribution

It proposes ε-strong security as a new security measure and shows that a variant of the geometric strategy can attain this level of security for the problem.

Findings

The geometric strategy can be adapted to achieve ε-strong security.

ε-strong security can be made arbitrarily close to perfect security.

The approach balances information sharing and security in card communication.

Abstract

In the Russian cards problem, Alice, Bob and Cath draw $a$, $b$ and $c$ cards, respectively, from a publicly known deck. Alice and Bob must then communicate their cards to each other without Cath learning who holds a single card. Solutions in the literature provide weak security, where Cath does not know with certainty who holds each card that is not hers, or perfect security, where Cath learns no probabilistic information about who holds any given card from Alice and Bob's exchange. We propose an intermediate notion, which we call $\varepsilon$-strong security, where the probabilities perceived by Cath may only change by a factor of $\varepsilon$. We then show that a mild variant of the so-called geometric strategy gives $\varepsilon$-strong safety for arbitrarily small $\varepsilon$ and appropriately chosen values of $a,b,c$.