# Super-Activation as a Unique Feature of Secure Communication in Malicious Environments

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## Abstract

**:**

## 1. Introduction

#### Notation

## 2. Arbitrarily Varying Wiretap Channels

#### 2.1. System Model

**Definition 1.**

**Definition 2.**

**Definition 3.**

#### 2.2. Code Concepts

#### 2.2.1. Unassisted Codes

**Definition 4.**

**Remark 1.**

**Definition 5.**

#### 2.2.2. CR-Assisted Codes

**Definition 6.**

**Remark 2.**

#### 2.3. Capacity Results

**Theorem 1**

**Theorem 2**

- If $\mathfrak{W}$ is symmetrizable, then ${C}_{S}(\mathfrak{W},\mathfrak{V})=0$.
- If $\mathfrak{W}$ is non-symmetrizable, then ${C}_{S}(\mathfrak{W},\mathfrak{V})={C}_{S,\text{CR}}(\mathfrak{W},\mathfrak{V})$.

**Theorem 3**

**Theorem 4**

- The AVWC $(\mathfrak{W},\mathfrak{V})$ is a discontinuity point of ${C}_{S}(\mathfrak{W},\mathfrak{V})$ if and only if the following holds: First, ${C}_{S,\text{CR}}(\mathfrak{W},\mathfrak{V})>0$, and second, $F(\mathfrak{W})=0$ but for every $\u03f5>0$ there is a finite ${\mathfrak{W}}^{\prime}$ with $D(\mathfrak{W},{\mathfrak{W}}^{\prime})\le \u03f5$ and $F({\mathfrak{W}}^{\prime})>0$.
- If ${C}_{S}(\mathfrak{W},\mathfrak{V})$ is discontinuous in the point $(\mathfrak{W},\mathfrak{V})$ then it is discontinuous for all ${\mathfrak{V}}^{\prime}$ for which ${C}_{S,\text{CR}}(\mathfrak{W},{\mathfrak{V}}^{\prime})>0$.

## 3. Super-Activation and Robustness

#### 3.1. Secure Communication over Orthogonal AVWCs

#### 3.2. Super-Activation of Orthogonal AVWCs

**Theorem 5**

- If ${C}_{S}({\mathfrak{W}}_{1},{\mathfrak{V}}_{1})={C}_{S}({\mathfrak{W}}_{2},{\mathfrak{V}}_{2})=0$, then$${C}_{S}({\mathfrak{W}}_{1}\otimes {\mathfrak{W}}_{2},{\mathfrak{V}}_{1}\otimes {\mathfrak{V}}_{2})>0$$$${C}_{S}({\mathfrak{W}}_{1}\otimes {\mathfrak{W}}_{2},{\mathfrak{V}}_{1}\otimes {\mathfrak{V}}_{2})={C}_{S,\text{CR}}({\mathfrak{W}}_{1}\otimes {\mathfrak{W}}_{2},{\mathfrak{V}}_{1}\otimes {\mathfrak{V}}_{2}).$$
- If ${C}_{S,\text{CR}}$ shows no super-activation for $({\mathfrak{W}}_{1},{\mathfrak{V}}_{1})$ and $({\mathfrak{W}}_{2},{\mathfrak{V}}_{2})$, then super-activation of ${C}_{S}$ can only happen if ${\mathfrak{W}}_{1}$ is non-symmetrizable and ${\mathfrak{W}}_{2}$ is symmetrizable and ${C}_{S,\text{CR}}({\mathfrak{W}}_{1},{\mathfrak{V}}_{1})=0$ and ${C}_{S,\text{CR}}({\mathfrak{W}}_{2},{\mathfrak{V}}_{2})>0$. The statement is independent of the specific labeling.
- There exist AVWCs that exhibit the behavior described by the second property.

**Theorem 6.**

**Proof.**

**Corollary 1.**

**Proof.**

**Theorem 7.**

**Proof.**

**Example 1.**

**Remark 3.**

**Theorem 8.**

**Proof.**

**Remark 4.**

**Remark 5.**

**Example 2.**

## 4. Communication over Orthogonal AVCs

#### 4.1. Capacity Results

**Theorem 9**

**Theorem 10**

#### 4.2. Additivity of CR-Assisted Capacity

**Theorem 11.**

**Proof.**

#### 4.3. Super-Additivity of Unassisted Capacity

**Proposition 1.**

**Proof.**

**Proposition 2.**

**Proof.**

**Corollary 2.**

**Theorem 12.**

**Proof.**

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Arbitrarily varying wiretap channel. The transmitter encodes the message M into the codeword ${X}^{n}=E(M)$ and transmits it over the AVWC to the legitimate receiver, which has to decode its intended message $\widehat{M}=\phi ({Y}_{{s}^{n}}^{n})$ for any state sequence ${s}^{n}\in {\mathcal{S}}^{n}$. At the same time, the eavesdropper must be kept ignorant of M by requiring ${max}_{{s}^{n}\in {\mathcal{S}}^{n}}I(M;{Z}_{{s}^{n}}^{n})\le {\delta}_{n}$.

**Figure 2.**CR is available to all users including the eavesdropper. The transmitter and receiver can adapt their encoder and decoder according to the actual CR realization $\gamma \in {\mathcal{G}}_{n}$.

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**MDPI and ACS Style**

Schaefer, R.F.; Boche, H.; Poor, H.V.
Super-Activation as a Unique Feature of Secure Communication in Malicious Environments. *Information* **2016**, *7*, 24.
https://doi.org/10.3390/info7020024

**AMA Style**

Schaefer RF, Boche H, Poor HV.
Super-Activation as a Unique Feature of Secure Communication in Malicious Environments. *Information*. 2016; 7(2):24.
https://doi.org/10.3390/info7020024

**Chicago/Turabian Style**

Schaefer, Rafael F., Holger Boche, and H. Vincent Poor.
2016. "Super-Activation as a Unique Feature of Secure Communication in Malicious Environments" *Information* 7, no. 2: 24.
https://doi.org/10.3390/info7020024