Polytope Codes Against Adversaries in Networks

TL;DR

This paper introduces Polytope Codes, a new class of codes designed to secure network communication against adversaries controlling unknown nodes, achieving optimal bounds in certain network classes.

Contribution

The paper proposes Polytope Codes for network security against node adversaries, demonstrating their effectiveness and establishing bounds in planar networks.

Findings

Polytope Codes achieve a cut-set bound in some planar networks.

Linear codes are insufficient against node adversaries.

A tighter bound is identified for specific network examples.

Abstract

Network coding is studied when an adversary controls a subset of nodes in the network of limited quantity but unknown location. This problem is shown to be more difficult than when the adversary controls a given number of edges in the network, in that linear codes are insufficient. To solve the node problem, the class of Polytope Codes is introduced. Polytope Codes are constant composition codes operating over bounded polytopes in integer vector fields. The polytope structure creates additional complexity, but it induces properties on marginal distributions of code vectors so that validities of codewords can be checked by internal nodes of the network. It is shown that Polytope Codes achieve a cut-set bound for a class of planar networks. It is also shown that this cut-set bound is not always tight, and a tighter bound is given for an example network.