# Integrating Classical Preprocessing into an Optical Encryption Scheme

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## Abstract

**:**

## 1. Introduction

#### Our Contribution

## 2. Background and Tools

#### 2.1. The AlphaEta Protocol

- ∘
- $\langle n\rangle $: average number of photons per pulse
- ∘
- $\beta $: number of bases used
- ∘
- s: number of pulses sent in one round of the protocol

#### 2.2. All-or-Nothing Transforms

- $\varphi $ is a bijection.
- If any $s-\ell $ of the s output values ${y}_{1},\cdots ,{y}_{s}$ are fixed, then any ℓ of the input values ${x}_{i}$ ($1\le i\le s$) are completely undetermined, in an information-theoretic sense.

**Definition**

**1.**

- 1.
- $H({Y}_{1},\cdots ,{Y}_{s}|{X}_{1},\cdots ,{X}_{s})=0$,
- 2.
- $H({X}_{1},\cdots ,{X}_{s}|{Y}_{1},\cdots ,{Y}_{s})=0$
- 3.
- For all $\mathcal{X}\subseteq \{{X}_{1},\cdots ,{X}_{s}\}$ with $\left|\mathcal{X}\right|=\ell $, and for all $\mathcal{Y}\subseteq \{{Y}_{1},\cdots ,{Y}_{s}\}$ with $\left|\mathcal{Y}\right|=\ell $, it holds that$$H\left(\mathcal{X}\right|\{{Y}_{1},\cdots ,{Y}_{s}\}\backslash \mathcal{Y})=H\left(\mathcal{X}\right).$$

**Definition**

**2.**

**Theorem**

**1.**

## 3. Results

#### 3.1. Symmetric-key Encryption Using Mesoscopic Coherent States

**Definition**

**3.**

`KeyGen`: Given a key length, outputs a corresponding secret key k.`Enc`: Given a plaintext m and secret key k, it outputs a ciphertext c, consisting of a sequence of coherent states and a bitstring:$$c=(|{\psi}_{1}\rangle ,\cdots ,|{\psi}_{j}\rangle ,{c}_{1},\cdots ,{c}_{\ell})$$`Dec`: This process consists of two phases. Given a ciphertext c and a secret key k, the sequence of coherent states in c is measured in the first phase. Now c can be considered a classical bitstring when entering the second phase of the decryption. The final output of the algorithm is the plaintext m.

**Remark**

**1.**

**Definition**

**4.**

**Definition**

**5.**

**Remark**

**2.**

**Definition**

**6.**

#### 3.2. A Hybrid Construction

#### 3.2.1. Description and Design Rationale

**Definition**

**7.**

- $H({Y}_{1},\cdots ,{Y}_{s}|{X}_{1},\cdots ,{X}_{s})=0$
- $H({X}_{1},\cdots ,{X}_{s}|{Y}_{1},\cdots ,{Y}_{s})=0$
- For all i such that $1\le i\le s$, $H\left({X}_{i}\right|{Y}_{2},\cdots ,{Y}_{s})=H\left({X}_{i}\right)$.

**Definition**

**8.**

- $H({Y}_{1},\cdots ,{Y}_{s}|{X}_{1},\cdots ,{X}_{s})=0$
- $H({X}_{1},\cdots ,{X}_{s}|{Y}_{1},\cdots ,{Y}_{s})=0$
- Let $\mathcal{Y}=\{{Y}_{\upsilon}:\upsilon \in {\rm Y}\}$ represent the collection of hidden bits. For all i such that $1\le i\le s$, it holds that$$H({X}_{i}|\{{Y}_{1},\cdots ,{Y}_{s}\}\backslash \mathcal{Y})=H\left({X}_{i}\right).$$

**Proposition**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.2.2. Security Analysis

**Proposition**

**2.**

**Proof.**

**Remark**

**5.**

**Theorem**

**2.**

**Proof.**

#### 3.3. Forward Security

**Theorem**

**3.**

**Proof.**

## 4. Discussion

#### 4.1. Integrating Classical Authenticated Encryption

#### 4.2. Choosing Parameters

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**How the message is sent using the hybrid construction with authenticated encryption scheme; AEAD represents the application of ChaCha20 and Poly1305.

**Figure 6.**A symmetric-key encryption scheme using mesoscopic coherent states with incorporated AEAD.

r | $\frac{\mathit{r}-1}{2}$ | $\mathit{\delta}$ | ${\mathit{\delta}}^{\prime}$ |
---|---|---|---|

3 | 1 | ${10}^{-9}$ | $3.84\times {10}^{-16}$ |

${10}^{-5}$ | $3.84\times {10}^{-8}$ | ||

${10}^{-1}$ | $0.9736$ | ||

7 | 3 | ${10}^{-9}$ | $4.48\times {10}^{-33}$ |

${10}^{-5}$ | $4.48\times {10}^{-17}$ | ||

${10}^{-1}$ | $0.2951$ | ||

101 | 50 | ${10}^{-9}$ | $2.56\times {10}^{-428}$ |

${10}^{-5}$ | $2.56\times {10}^{-224}$ | ||

${10}^{-1}$ | $1.47\times {10}^{-22}$ |

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**MDPI and ACS Style**

Pham, H.; Steinwandt, R.; Suárez Corona, A. Integrating Classical Preprocessing into an Optical Encryption Scheme. *Entropy* **2019**, *21*, 872.
https://doi.org/10.3390/e21090872

**AMA Style**

Pham H, Steinwandt R, Suárez Corona A. Integrating Classical Preprocessing into an Optical Encryption Scheme. *Entropy*. 2019; 21(9):872.
https://doi.org/10.3390/e21090872

**Chicago/Turabian Style**

Pham, Hai, Rainer Steinwandt, and Adriana Suárez Corona. 2019. "Integrating Classical Preprocessing into an Optical Encryption Scheme" *Entropy* 21, no. 9: 872.
https://doi.org/10.3390/e21090872