# Separable Reversible Data Hiding in Encrypted Signals with Public Key Cryptography

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Paillier Cryptosystem

^{λ}mod n

^{2}), n) = 1, and $L\left(x\right)=\frac{x-1}{n}$. Finally, (n, g) is the public key and (λ) is the private key.

_{pk}(m, r) = g

^{m}r

^{n}mod n

^{2},

_{pk}(m

_{1}, r

_{1}) and E

_{pk}(m

_{2}, r

_{2}), the two functions are additively homomorphic on ${\mathbb{Z}}_{n}$:

_{sk}(E

_{pk}(m

_{1}, r

_{1}) × E

_{pk}(m

_{2}, r

_{2})) = m

_{1}+ m

_{2}.

_{sk}(E

_{pk}(m

_{1}, r

_{1})

^{k}) = km

_{1},

_{sk}(E

_{pk}(m

_{1}, r

_{1}) × g

^{m}

^{2}mod n

^{2}) = m

_{1}+ m

_{2},

## 3. Proposed Scheme

#### 3.1. Image Encryption

_{i}

_{, j}denotes the grayscale value at the coordinate of (i, j), where 1 ≤ i ≤ H, 1 ≤ j ≤ W, and 0 ≤ x

_{i}

_{, j}≤ 255. The detailed procedure is listed below.

- Step 1.
- For each pixel x
_{i}_{, j}, convert x_{i}_{, j}to x^{1}_{i}_{, j}and x^{2}_{i}_{, j}, where x_{i}_{, j}= x^{1}_{i}_{, j}+ x^{2}_{i}_{, j}. - Step 2.
- Choose a random integer r
_{1}∈ ${\mathbb{Z}}_{n}^{\ast}$, and then computes the encryption function E_{pk}(x^{1}_{i}_{, j}, r_{1}) with a public key by Equation (1). - Step 3.
- Choose a random integer r
_{2}∈ ${\mathbb{Z}}_{n}^{\ast}$, and then computes the encryption function E_{pk}(x^{2}_{i}_{, j}, r_{2}) with a public key by Equation (1) so as to meet that E_{pk}(x^{1}_{i}_{, j}, r_{1}) ≠ E_{pk}(x^{2}_{i}_{, j}, r_{2}). - Step 4.
- All the encrypted units comprise the encrypted signal.

#### 3.2. Data Embedding

_{i}= (EU

^{1}

_{i}, EU

^{2}

_{i}), where 1 ≤ i ≤ W × H.

- Step 1.
- Construct a non-repeat random embedding sequence using the data-hiding key.
- Step 2.
- Embed a secret bit into an encrypted unit EU
_{i}according to the embedding sequence. If the secret bit is 1 and EU^{1}_{i}< EU^{2}_{i}, EU^{1}_{i}swaps EU^{2}_{i}. - Step 3.
- If the secret bit is 0 and EU
^{1}_{i}> EU^{2}_{i}, EU^{1}_{i}swaps EU^{2}_{i}. - Step 4.
- Generate a marked encrypted signal when all bits are embedded.

#### 3.3. Data Extraction

_{i}= (MEU

^{1}

_{i}, MEU

^{2}

_{i}), where 1 ≤ i ≤ W × H. If the receiver has only a data-hiding key, the data extraction procedure is listed below.

- Step 1.
- Step 1. Construct a non-repeat random embedding sequence using the data-hiding key.
- Step 2.
- Extract a secret bit from a marked encrypted unit MEU
_{i}according to the embedding sequence. If MEU^{1}_{i}> MEU^{2}_{i}, the extracted bit is 1. - Step 3.
- If MEU
^{1}_{i}< MEU^{2}_{i}, the extracted bit is 0. - Step 4.
- Obtain the embedded data when all the bits are extracted.

#### 3.4. Image Recovery

_{i}= (MEU

^{1}

_{i}, MEU

^{2}

_{i}), where 1 ≤ i ≤ W × H. We assume that the receiver has only a private key. The image recovery procedure is listed below.

- Step 1.
- Decrypt the marked encrypted unit MEU
_{i}using the private key by:D_{sk}((MEU^{1}_{i}× MEU^{2}_{i}) mod n^{2}) = x^{1}_{i}_{, j}+ x^{2}_{i}_{, j}= x_{i}_{, j}. - Step 2.
- Recover the original image when all marked encrypted units are decrypted.

_{i}

_{, j}= x

^{1}

_{i}

_{, j}+ x

^{2}

_{i}

_{, j}= 100 + 68 = 168, and set the secret bit as 0, two primes as p = 17 and q = 19. Thus, (323, 324) is the public key, and (144) is the private key. Compute the encrypted unit EU

_{i}= (EU

^{1}

_{i}, EU

^{2}

_{i}) = (E

_{pk}(100, 7), E

_{pk}(68, 11)) = (74,871, 34,549). In the data embedding phase, because the secret bit is 0 and EU

^{1}

_{i}> EU

^{2}

_{i}, EU

^{1}

_{i}swaps EU

^{2}

_{i}. Therefore, the marked encrypted unit MEU

_{i}= (EU

^{2}

_{i}, EU

^{1}

_{i}) = (34,549, 74,871). At the receiver side, the receiver who has only data-hiding key can extract secret bit 0 because MEU

^{1}

_{i}< MEU

^{2}

_{i}. However, if the receiver has only a private key, he can decrypt the marked encrypted unit MEU

_{i}to obtain the original pixel by computing D

_{sk}((MEU

^{1}

_{i}× MEU

^{2}

_{i}) mod n

^{2}) = D

_{sk}((34,549 × 74,871) mod 323

^{2}) = D

_{sk}(89,282) = 168.

## 4. Experimental Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Framework VRAE [4].

**Figure 2.**Framework RRBE [5].

**Figure 3.**Non-separable reversible data hiding in encrypted image [1].

**Figure 4.**Three cases at the receiver side of separable reversible data hiding [8].

Original Image | Embedding Rate (bpp) | PSNR (dB) |
---|---|---|

Lena | 1 | +∞ |

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**MDPI and ACS Style**

Tai, W.-L.; Chang, Y.-F.
Separable Reversible Data Hiding in Encrypted Signals with Public Key Cryptography. *Symmetry* **2018**, *10*, 23.
https://doi.org/10.3390/sym10010023

**AMA Style**

Tai W-L, Chang Y-F.
Separable Reversible Data Hiding in Encrypted Signals with Public Key Cryptography. *Symmetry*. 2018; 10(1):23.
https://doi.org/10.3390/sym10010023

**Chicago/Turabian Style**

Tai, Wei-Liang, and Ya-Fen Chang.
2018. "Separable Reversible Data Hiding in Encrypted Signals with Public Key Cryptography" *Symmetry* 10, no. 1: 23.
https://doi.org/10.3390/sym10010023