# Secret Sharing and Shared Information

## TL;DR

This paper explores the use of secret sharing schemes as models for information decomposition, revealing limitations in current frameworks and suggesting the need for lattice extensions to better capture redundancy.

## Contribution

It identifies the incompleteness of the partial information lattice and proposes that extending it can improve the understanding of information redundancy.

## Key findings

- Partial information lattice may be incomplete.
- Negative partial information terms arise in certain decompositions.
- Extensions to the lattice could better represent higher-order redundancy.

## Abstract

Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.