Common information in well-mixing graphs and applications to information-theoretic cryptography

TL;DR

This paper explores how mixing properties of bipartite graphs influence the mutual information shared across edges, revealing limitations in information extraction and implications for one-shot cryptographic protocols.

Contribution

It establishes a link between graph mixing properties and mutual information impossibility, applying these insights to one-shot cryptography and communication complexity.

Findings

Mixing properties imply mutual information extraction impossibility.

Communication complexity in one-shot secret key agreement can be uneven.

Certain graph structures restrict information sharing in cryptographic tasks.

Abstract

We study the connection between mixing properties for bipartite graphs and materialization of the mutual information in one-shot settings. We show that mixing properties of a graph imply impossibility to extract the mutual information shared by the ends of an edge randomly sampled in the graph. We apply these impossibility results to some questions motivated by information-theoretic cryptography. In particular, we show that communication complexity of a secret key agreement in one-shot setting is inherently uneven: for some inputs, almost all communication complexity inevitably falls on only one party.