# Secret Sharing and Shared Information

## Abstract

**:**

## 1. Introduction

## 2. Perfect Secret Sharing Schemes

**Definition**

**1.**

- $H({X}_{A},S)=H\left({X}_{A}\right)$, whenever $A\in \mathcal{A}$.

- $H({X}_{A},S)=H\left({X}_{A}\right)+H\left(S\right)$, whenever $A\notin \mathcal{A}$.

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

**Proof.**

## 3. Information Decompositions of Secret Sharing Schemes

**Definition**

**3.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**4.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**5.**

**Proof.**

## 4. Incompatibility with Local Positivity

**Theorem**

**2.**

**Proof.**

- ${I}_{\cap}\left(S;{X}_{1},\left({X}_{2}{X}_{3}\right)\right)={I}_{\cap}\left(S;{X}_{1},\left({X}_{1}{X}_{2}{X}_{3}\right)\right)={I}_{\cap}(S;{X}_{1})=1\mathrm{bit}$, since ${X}_{1}$ is a function of $({X}_{2},{X}_{3})$ and by the monotonicity axiom.
- ${I}_{\cap}(S;{X}_{1},{X}_{2})={I}_{\cap}\left(\left({X}_{1}{X}_{2}{X}_{3}\right);{X}_{1},{X}_{2}\right)={I}_{\cap}\left(\left({X}_{1}{X}_{2}\right);{X}_{1},{X}_{2}\right)=0$ by Lemma 5.

**Remark**

**1.**

## 5. Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A. Combined Secret Sharing Properties for Small k

**Lemma**

**A1.**

**Proof.**

**Proposition**

**A1.**

**Proof.**

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**Figure 1.**The partial information lattice for $n=3$. Each node is indexed by an antichain. The values (in bit) of the shared information in the XOR example from the proof of Theorem 2 according to the pairwise secret sharing property are given after the colon.

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Rauh, J.
Secret Sharing and Shared Information. *Entropy* **2017**, *19*, 601.
https://doi.org/10.3390/e19110601

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Rauh J.
Secret Sharing and Shared Information. *Entropy*. 2017; 19(11):601.
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Rauh, Johannes.
2017. "Secret Sharing and Shared Information" *Entropy* 19, no. 11: 601.
https://doi.org/10.3390/e19110601