# Generalized Scheme Based on Octagon-Shaped Shell for Data Hiding in Steganographic Applications

## Abstract

**:**

## 1. Introduction

_{1},x

_{2}) = (1 × x

_{1}+ 2 × x

_{2}) mod 5, we can construct a hypercube as shown in Figure 1. Suppose (x

_{1},x

_{2}) = (3,2) is a pixel pair of the cover image. If the to-be-embedded 5-ary secret digit is 0, 1, 2, 3, or 4, then we choose stego-pixels (3,1), (2,2), (3,2), (4,2), or (3,3), respectively.

^{2}+ 2k + 1)-ary numeral system secret digit and increased the payload up to 2.68 bpp (k = 4) [8]. Kim et al. proposed the 2-EMD and EMD-2 schemes based on a log

_{2}(2 (5n − 7) + 1)-ary numeral system secret digit and achieved a maximum payload of log

_{2}(10) bpp (about 3.22 bpp) [9]. Kieu and Chang improved the EMD scheme by using eight modification directions on a square-shaped shell of size s × s and attained a maximum payload of up to 4.5 bpp (s = 23) [10]. Kim used a codebook to embed a (2

^{n}

^{+x}− 1)-ary numeral system secret digit and achieved a maximum payload of up to 2.48 bpp [11]. Liu et al. extended the EMD scheme to realize a payload of 2.5 bpp [12]. Leng and Tseng extended the EMD scheme on a three-dimensional (3D)-hypercube sized w × w × w and generalized the EMD scheme on an n-dimensional hypercube with a maximum payload of up to 4.75 bpp (w = 27) [13].

## 2. Related Works

_{1},x

_{2}), the location of M (x

_{1},x

_{2}) determines the to-be-embedded secret digits. To embed the secret digit, s, into the pixel pair, (x

_{1},x

_{2}), the pixel pair is first located in the reference matrix, M, as M (x

_{1},x

_{2}), and the element, M (x

_{1}′,x

_{2}′), which has the shortest distance with M (x

_{1},x

_{2}) and M (x

_{1}′,x

_{2}′) = s, is searched. Subsequently, the original pixel pair, (x

_{1},x

_{2}), is modified to (x

_{1}′,x

_{2}′). In the extraction procedure, the secret digit is extracted correctly from the stego pixel pair that is mapped to the element, M(x

_{1}′,x

_{2}′), in the reference matrix, M.

#### 2.1. Scheme of Chang et al.

_{2}= 7. The cover pixel pair, (3,4), is mapped to M (3,4) = 3 in the reference matrix, M. Then, the candidate stego pixel pairs, M (1,5) = 7, M (2,2) = 7, M (4,5) = 7, and M (5,3) = 7, choose (4,5) as the stego pixel pair to ensure the lowest distortion. In the extraction procedure, if M (4,5) = 7 = (111)

_{2}; then, the secret digits are extracted correctly.

#### 2.2. Scheme of Kurup et al.

_{2}= 11. The cover pixel pair, (4,2), is mapped to M (4,2) = 3 in the reference matrix, M. Then, the candidate stego pixel pairs are M (2,2) = 11, M (5,5) = 11, and M (6,1) = 11, and (2,2) is selected as the stego pixel pair to ensure the lowest distortion. In the extraction procedure, if M (2,2) = 11 = (1011)

_{2}, then the secret digit can be extracted correctly.

#### 2.3. Scheme of Leng

^{2}− 4 = 2

^{k}

_{2}= 23. The cover pixel pair, (6,7), is mapped to M (6,7) = 8 in the reference matrix, M. Then, the candidate stego pixel pairs are M (3,6) = 23, M(6,10) = 23, and M (9,4) = 23, and (3,6) is selected as the stego pixel pair to ensure the lowest distortion. In the extraction procedure, if M (3,6) = 23 = (10111)

_{2}, then the secret digits can be extracted correctly.

## 3. The Proposed Method

^{w}different digits ranging from 0 to 2

^{w}− 1 digits. Then, for each pixel pair, the proposed method embeds w secret bits. The construction of reference matrix M, sized 256 × 256, is based on the following three rules. Let M = (a

_{i}

_{,j})

_{256}

_{×256}; then, element a

_{0,0}can choose any number from the range [0,2

^{w}−1] and serve as a key. The value difference between two adjacent columns in the same row is set as 1, and the value between the two adjacent rows is set alternately as d

_{0}, d

_{1}, d

_{2}, …, d

_{m}

_{−1}, where d

_{i}is defined as follows:

d_{0 }= a_{0,0}, d_{i} = a_{i}_{+1,0} − a_{i}_{,0} | $0\le i\le m-1$ |

${d}_{i}={d}_{i-1}+\left(n-2\left|\left(k+1\right)-i\right|+1\right)$ | $1\le i\le k$ |

${d}_{i}={d}_{i-1}+\left(n-2\left|\left(m-1-k\right)-i\right|+1\right)$ | $m-k\le i\le m-1$ |

d_{i} = d_{i}_{−1} + n | $k+1\le i\le m-1-k$ |

_{1},x

_{2}), if the to-be-embedded secret digits are s, then the element M (x

_{1}′,x

_{2}′), which has the shortest distance with M (x

_{1},x

_{2}) and M (x

_{1}′,x

_{2}′) = s, is searched. In the extraction procedure, the secret digit is extracted correctly from the stego pixel pair, which is mapped to the element, M (x

_{1}′,x

_{2}′), in the reference matrix, M.

## 4. Results and Discussion

_{i}

_{,j}and x

_{i}

_{,j}′ are the pixel values of the cover image and stego-image, respectively. Table 2 present the PSNR values of the proposed method as a function of the payloads (w = 3, 4, 5, 6, 7, and 8).

_{1},x

_{2}) is located in the border area of the reference matrix, it causes a large distortion when it searches the element, M (x

_{1}′,x

_{2}′) = s. When w = 8, the payload is 4 bpp and there are eleven cases. Similarly, the cases (m,n,k) = (4,65,1) and (m,n,k) = (5,52,1) are two poor cases. However, the case (m,n,k) = (14,20,3) achieves a high payload (4 bpp) with a good visual stego-image quality (34.70 dB).

## 5. Conclusions

_{1},x

_{2}) is located in the border area of the reference matrix, it causes a large distortion when it searches the element M(x

_{1}′,x

_{2}′) = s. In the future, we plan to improve Equation (2) so that the number of digits contains in an octagon-shape shells is m × n instead of 2

^{w}.

## Conflicts of Interest

## References

- Tuner, L.F. Digital Data Security System. Patent IPN WO 89/08915, 21 September 1989. [Google Scholar]
- Chan, C.-K.; Cheng, L. Hiding data in images by simple LSB substitution. Pattern Recognit.
**2004**, 37, 469–474. [Google Scholar] [CrossRef] - Thien, C.-C.; Lin, J.-C. A simple and high-hiding capacity method for hiding digit-by-digit data in images based on modulus function. Pattern Recognit.
**2003**, 36, 2875–2881. [Google Scholar] [CrossRef] - Li, X.; Yang, B.; Cheng, D.; Zeng, T. A generalization of LSB matching. IEEE Signal Proc. Lett.
**2009**, 16, 69–72. [Google Scholar] - Mielikainen, J. LSB matching revisited. IEEE Signal Proc. Lett.
**2006**, 13, 285–287. [Google Scholar] [CrossRef] - Fridrich, J.; Goljan, M.; Du, R. Reliable Detection of LSB Steganography in Color and Grayscale Images. In Proceedings of the 2001 Workshop on Multimedia and Security New Challenges, Ottawa, ON, Canada, 5 October 2001. [Google Scholar]
- Zhang, X.; Wang, S. Efficient steganographic embedding by exploiting modification direction. IEEE Commun. Lett.
**2006**, 10, 781–783. [Google Scholar] [CrossRef] - Chao, R.-M.; Wu, H.-C.; Lee, C.-C.; Chu, Y.-P. A novel image data hiding scheme with diamond encoding. EURASIP J. Inf. Sec.
**2009**, 2009, 1–9. [Google Scholar] [CrossRef] - Kim, H.; Kim, C.; Choi, Y.; Wang, S.; Zhang, X. Improved modification direction methods. Comput. Math. Appl.
**2010**, 60, 319–325. [Google Scholar] [CrossRef] [Green Version] - Kieu, T.D.; Chang, C.-C. A steganographic scheme by fully exploiting modification directions. Expert Syst. Appl.
**2011**, 38, 10648–10657. [Google Scholar] [CrossRef] - Kim, C. Data hiding by an improved exploiting modification direction. Multimed. Tools Appl.
**2012**, 69, 569–584. [Google Scholar] [CrossRef] - Liu, Y.; Chang, C.-C.; Huang, P.-C. Extended Exploiting-Modification-Direction Data Hiding with High Capacity. In Proceedings of the International Conference on Video and Image Processing—ICVIP 2017, Singapore, 27–29 December 2017. [Google Scholar]
- Leng, H.-S.; Tseng, H.-W. Generalize the EMD scheme on an n-dimensional hypercube with maximum payload. Multimed. Tools Appl.
**2019**, 1–15. [Google Scholar] [CrossRef] - Chang, C.C.; Liu, Y.; Nguyen, T.S. A Novel Turtle Shell Based Scheme for Data Hiding. In Proceedings of the 2014 Tenth International Conference on Intelligent Information Hiding and Multimedia Signal Processing, Kitakyushu, Japan, 27–29 August 2014. [Google Scholar]
- Kurup, S.; Rodrigues, A.; Bhise, A. Data hiding scheme based on octagon shaped shell. In Proceedings of the 2015 International Conference on Advances in Computing, Communications and Informatics (ICACCI), Kerala, India, 10–13 August 2015. [Google Scholar]
- Leng, H.-S. Data Hiding Scheme Based on Regular Octagon-Shaped Shells. In Proceedings of the Advances in Intelligent Information Hiding and Multimedia Signal Processing Smart Innovation, Systems and Technologies, Kaohsiung, Taiwan, 21–23 November 2017. [Google Scholar]
- Leng, H.-S.; Tseng, H.-W. Maximizing the payload of the octagon-shaped shell-based data hiding scheme. In Proceedings of the 2017 IEEE 8th International Conference on Awareness Science and Technology (iCAST), Taichung, Taiwan, 8 November 2017. [Google Scholar]

**Figure 7.**Reference matrix (

**a**) of the proposed method with (m,n,k,w) = (3,4,1,3) (

**b**) reconstructed by hexagon-shaped shells to compare with the scheme of Chang et al.

**Figure 8.**Six 512 × 512 grayscale images (

**a**) Tiffany (

**b**) Baboon (

**c**) Lena (

**d**) Jet (

**e**) Scene (

**f**) Peppers.

k = 1 | k = 2 | k = 3 | k = 4 | k = 6 | k = 7 | k = 8 | |
---|---|---|---|---|---|---|---|

(m,n,w) | (3,4,3)^{1},(3,12,5), (3,44,7), (4,5,4) ^{2},(4,9,5), (4,17,6), (4,33,7), (4,65,8), (5,52,8), (6,6,5) ^{3},(6,22,7), (10,26,8), (11,12,7) ^{4},(13,20,8) | (5,28,7), (7,20,7), (10,14,7) | (7,8,5), (7,40,8), (8,11,6), (8,19,7), (8,35,8), (10,28,8), (14,20,8) | (12,14,7) | (17,20,8) | (15,16,7), (16,23,8) | (20,20,8) |

solutions | 14 | 3 | 7 | 1 | 1 | 2 | 1 |

^{1}Scheme of Chang et al. is the case of (m,n,k,w) = (3,4,1,3).

^{2}Scheme of Kurup et al. is the case of (m,n,k,w) = (4,5,1,4).

^{3}Scheme of Leng is the case of (m,n,k,w) = (6,6,1,5).

^{4}Scheme of Leng and Tseng is the case of (m,n,k,w) = (11,12,1,7).

(m,n,k) | w = 3 | (m,n,k) | w = 4 | (m,n,k) | w = 5 | (m,n,k) | w = 6 | (m,n,k) | w = 7 | (m,n,k) | w = 8 |
---|---|---|---|---|---|---|---|---|---|---|---|

(3,4,1) | 49.74 | (4,5,4) | 46.82 | (3,12,1) | 41.34 | (4,17,1) | 37.95 | (3,44,1) | $\underset{=}{29.05}$ | (4,65,1) | $\underset{=}{25.05}$ |

(4,9,1) | 43.14 | (8,11,3) | 40.71 | (4,33,1) | 31.83 | (5,52,1) | $\underset{=}{27.28}$ | ||||

(6,6,1) | 43.94 | (6,22,1) | 35.17 | (10,26,1) | 33.05 | ||||||

(7,8,3) | 43.31 | (11,12,1) | 37.83 | (13,20,1) | 34.40 | ||||||

(5,28,2) | 34.02 | (7,40,3) | 30.79 | ||||||||

(7,20,2) | 36.49 | (8,35,3) | 31.96 | ||||||||

(10,14,2) | 37.77 | (10,28,3) | 33.45 | ||||||||

(8,19,3) | 37.23 | (14,20,3) | 34.70 | ||||||||

(12,14,4) | 37.61 | (17,20,6) | 34.51 | ||||||||

(15,16,7) | 36.49 | (16,23,7) | 34.53 | ||||||||

(20,20,8) | 33.79 |

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**MDPI and ACS Style**

Leng, H.-S.
Generalized Scheme Based on Octagon-Shaped Shell for Data Hiding in Steganographic Applications. *Symmetry* **2019**, *11*, 760.
https://doi.org/10.3390/sym11060760

**AMA Style**

Leng H-S.
Generalized Scheme Based on Octagon-Shaped Shell for Data Hiding in Steganographic Applications. *Symmetry*. 2019; 11(6):760.
https://doi.org/10.3390/sym11060760

**Chicago/Turabian Style**

Leng, Hui-Shih.
2019. "Generalized Scheme Based on Octagon-Shaped Shell for Data Hiding in Steganographic Applications" *Symmetry* 11, no. 6: 760.
https://doi.org/10.3390/sym11060760