An operational characterization of mutual information in algorithmic information theory

TL;DR

This paper links Kolmogorov complexity-based mutual information to the maximum secret key length achievable through probabilistic protocols with public communication, providing a precise operational interpretation.

Contribution

It establishes an operational characterization of mutual information in algorithmic information theory and determines the communication complexity for secret key agreement protocols.

Findings

Mutual information equals the length of the longest shared secret key up to logarithmic factors.

The maximum secret key length is tied to the complexity of the tuple minus minimal communication.

Protocols with below-threshold communication yield only very short secret keys.

Abstract

We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings $x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having $x$ and the complexity profile of the pair and the other one having $y$ and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For $\ell > 2$, the longest shared secret that can be established from a tuple of strings $(x_1, \ldots , x_\ell)$ by $\ell$ parties, each one having one component of the tuple and the complexity profile of the tuple, is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length, for protocols with public randomness. We also show that if the communication complexity drops below the established threshold, then only very short secret keys can be obtained.