Information Hiding Using Matroid Theory

TL;DR

This paper explores the use of matroid theory to analyze and optimize information hiding protocols, particularly in private information retrieval, by examining the structure of linear codes and their associated matroids.

Contribution

It introduces a matroid-theoretic framework to evaluate the capacity of linear codes for secret transmission and provides conditions for when secret sharing is impossible.

Findings

The efficiency of secret transmission protocols is linked to the derived matroid of the linear code.

Sufficient combinatorial conditions are identified for the impossibility of secret transmission.

Matroid properties can determine the maximum size of linear codes for secure communication.

Abstract

Inspired by problems in Private Information Retrieval, we consider the setting where two users need to establish a communication protocol to transmit a secret without revealing it to external observers. This is a question of how large a linear code can be, when it is required to agree with a prescribed code on a collection of coordinate sets. We show how the efficiency of such a protocol is determined by the derived matroid of the underlying linear communication code. Furthermore, we provide several sufficient combinatorial conditions for when no secret transmission is possible.