# An Approach for Security Enhancement of Certain Encryption Schemes Employing Error Correction Coding and Simulated Synchronization Errors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposal for a Security Enhanced Encryption

**M**. Encrypted form of

**m**is denoted by $\mathbf{c}\in {\{0,1\}}^{{n}^{\prime}}$ and we assume that it is a realization of the binary vector variable $\mathbf{C}$:

_{2}. The attacker does not know the specific realization of the “individual channel selection events”, i.e., they do not know which specific sub-channel bit is transmitted through, and which specific sub-channel each output symbol is received from.

## 3. Preliminaries and Background

#### 3.1. Entropy, Mutual Information, and Shannon Capacity

#### 3.2. Mutual Information and Capacity of the Deletion Channel with Fragmentation

#### 3.3. The Probability of Error and the Equivocation after a Noisy Channel

## 4. Security Evaluation of the Enhanced Encryption

#### 4.1. Security Notation

**Definition**

**1**

- 1.
- The adversary $\mathcal{A}$ chooses a pair of messages $({\mathbf{m}}_{0};{\mathbf{m}}_{1})$ of the same length n, and passes them on to the encryption system for encrypting.
- 2.
- A bit $b\phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}\{0,\phantom{\rule{-0.166667em}{0ex}}1\}$ is chosen uniformly at random, and only one of the two messages $({\mathbf{m}}_{0};{\mathbf{m}}_{1})$, precisely ${\mathbf{m}}_{b}$, is encrypted into ciphertext $\mathrm{Enc}\left({\mathbf{m}}_{b}\right)$ and returned to $\mathcal{A}$;
- 3.
- Upon observing $\mathrm{Enc}\left({\mathbf{m}}_{b}\right)$, and without knowledge of b, the adversary $\mathcal{A}$ outputs a bit ${b}_{0}$;
- 4.
- The experiment output is defined to be 1 if ${b}_{0}=b$, and 0 otherwise; if the experiment output is 1, denoted shortly as the event $(\mathcal{A}\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}1)$, we say that $\mathcal{A}$ has succeeded.

**Definition**

**2**

#### 4.2. Evaluation of the Security Gain

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 5. Notes on Implementation Issues

- Input: ${\mathbf{x}}^{\left(n\right)}={\left[{x}_{i}\right]}_{i=1}^{n}$, the parameter ${\Delta}^{*}<\Delta $
- set w = 1.
- do i = 1, n
- -
- if $w\le {\Delta}^{*}$${y}_{i}$ = ? and $w=w+1$ if ${\ell}_{i}\xb7{e}_{i}^{\left(1\right)}=1$ or ${\ell}_{i}\xb7{e}_{i}^{\left(2\right)}=1$${y}_{i}={x}_{i}$ otherwise
- -
- if $w>{\Delta}^{*}$${y}_{i}$ = ? if ${\ell}_{i}\xb7{e}_{i}^{\left(2\right)}=1$${y}_{i}={x}_{i}$ otherwise

- Output: ${\mathbf{y}}^{\left(n\right)}={\left[{y}_{i}\right]}_{i=1}^{n}$

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Model of encryption for cryptanalysis at the attacker’s side under known plaintext attack.

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**MDPI and ACS Style**

Mihaljević, M.J.; Wang, L.; Xu, S.
An Approach for Security Enhancement of Certain Encryption Schemes Employing Error Correction Coding and Simulated Synchronization Errors. *Entropy* **2022**, *24*, 406.
https://doi.org/10.3390/e24030406

**AMA Style**

Mihaljević MJ, Wang L, Xu S.
An Approach for Security Enhancement of Certain Encryption Schemes Employing Error Correction Coding and Simulated Synchronization Errors. *Entropy*. 2022; 24(3):406.
https://doi.org/10.3390/e24030406

**Chicago/Turabian Style**

Mihaljević, Miodrag J., Lianhai Wang, and Shujiang Xu.
2022. "An Approach for Security Enhancement of Certain Encryption Schemes Employing Error Correction Coding and Simulated Synchronization Errors" *Entropy* 24, no. 3: 406.
https://doi.org/10.3390/e24030406