# Image Encryption and Decryption Systems Using the Jigsaw Transform and the Iterative Finite Field Cosine Transform

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. Jigsaw Transform (JT)

#### 2.2. Finite Field Cosine Transform (FFCT)

## 3. Image Encryption and Decryption Systems Based on JT and FFCT

## 4. Numerical Experiments

#### 4.1. Statistical Analysis

#### 4.2. Entropy Analysis

#### 4.3. Key Space

#### 4.4. Differential Attack

#### 4.5. Key Sensitivity

#### 4.6. Computing Time

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Gu, Z.H.; Leger, J.R.; Lee, S.H. Optical computations of cosine transforms. Opt. Commun.
**1981**, 39, 137–142. [Google Scholar] [CrossRef] - Goodman, J.W. Introduction to Fourier Optics; McGraw-Hill: New York, NY, USA, 1996. [Google Scholar]
- Ozaktas, H.M.; Zalevsky, Z.; Kutay, M.A. The Fractional Fourier Transform: With Applications in Optics and Signal Processing; Wiley: Hoboken, NJ, USA, 2001. [Google Scholar]
- Rao, K.; Yip, P. Discrete Cosine Transform: Algorithms, Advantages, Applications; Academic Press: San Diego, CA, USA, 2014. [Google Scholar]
- Krikor, L.; Baba, S.; Arif, T.; Shaaban, Z. Image encryption using DCT and stream cipher. Eur. J. Sci. Res.
**2009**, 32, 47–57. [Google Scholar] - Liu, Z.; Xu, L.; Liu, T.; Chen, H.; Li, P.; Lin, C.; Liu, S. Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains. Opt. Commun.
**2011**, 284, 123–128. [Google Scholar] [CrossRef] - Pan, S.M.; Wen, R.H.; Zhou, Z.H.; Zhou, N.R. Optical multi-image encryption scheme based on discrete cosine transform and nonlinear fractional Mellin transform. Multimed. Tools Appl.
**2017**, 76, 2933–2953. [Google Scholar] [CrossRef] - Hua, Z.; Zhou, Y.; Huang, H. Cosine-transform-based chaotic system for image encryption. Inf. Sci.
**2019**, 480, 403–419. [Google Scholar] [CrossRef] - Kumar, S.; Panna, B.; Jha, R.K. Medical image encryption using fractional discrete cosine transform with chaotic function. Med. Biol. Eng. Comput.
**2019**, 57, 2517–2533. [Google Scholar] [CrossRef] [PubMed] - Schroeder, M. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity; Springer: Berlin, Germany, 2009. [Google Scholar]
- Bhattacharya, M.; Creutzburg, R.; Astola, J. Some historical notes on number theoretic transform. In Proceedings of the International TICS Workshop on Spectral Methods and Multirate Signal Processing, Vienna, Austria, 11–12 September 2004. [Google Scholar]
- Abdallah, E.E.; Hamza, A.B.; Bhattacharya, P. MPEG video watermarking using tensor singular value decomposition. In Proceedings of the International Conference Image Analysis and Recognition, Montreal, QC, Canada, 22–24 August 2007. [Google Scholar]
- Abdallah, E.E.; Hamza, A.B.; Bhattacharya, P. Video watermarking using wavelet transform and tensor algebra. Signal Image Video Process.
**2010**, 4, 233–245. [Google Scholar] [CrossRef] - Stoyanov, B.; Kordov, K. Novel image encryption scheme based on Chebyshev polynomial and duffing map. Sci. World J.
**2014**, 2014, 283639. [Google Scholar] [CrossRef] [PubMed] - Stoyanov, B.; Kordov, K. Image encryption using Chebyshev map and rotation equation. Entropy
**2015**, 17, 2117–2139. [Google Scholar] [CrossRef] - Lima, J.B.; de Souza, R.M.C. Finite field trigonometric transforms. Appl. Algebr. Eng. Commun.
**2011**, 22, 393–411. [Google Scholar] [CrossRef] - Lima, J.; Lima, E.; Madeiro, F. Image encryption based on the finite field cosine transform. Signal Process. Image Commun.
**2013**, 28, 1537–1547. [Google Scholar] [CrossRef] - Lima, J.; Madeiro, F.; Sales, F. Encryption of medical images based on the cosine number transform. Signal Process. Image Commun.
**2015**, 35, 1–8. [Google Scholar] [CrossRef] - Mikhail, M.; Abouelseoud, Y.; ElKobrosy, G. Two-phase image encryption scheme based on FFCT and fractals. Secur. Commun. Netw.
**2017**, 2017, 7367518. [Google Scholar] [CrossRef] - Millán, M.S.; Pérez-Cabré, E. Optical data encryption. In Optical and Digital Image Processing: Fundamentals and Applications; Cristóbal, G., Schelkens, P., Thienpont, H., Eds.; Wiley-VCH Verlag GmbH & Co.: Hoboken, NJ, USA, 2011; pp. 739–767. [Google Scholar]
- Millán, M.S.; Pérez-Cabré, E.; Vilardy, J.M. Nonlinear techniques for secure optical encryption and multifactor authentication. In Advanced Secure Optical Image Processing for Communications; Al Falou, A., Ed.; IOP Publishing: Bristol, UK, 2018; pp. 8-1–8-33. [Google Scholar]
- Muniraj, I.; Sheridan, J.T. Optical Encryption and Decryption; SPIE: Bellingham, WA, USA, 2019. [Google Scholar]
- Hennelly, B.; Sheridan, J.T. Optical image encryption by random shifting in fractional Fourier domains. Opt. Lett.
**2003**, 28, 269–271. [Google Scholar] [CrossRef][Green Version] - Vilardy, J.M.; Torres, C.O.; Mattos, L. Image encryption-decryption system based on Gyrator transform and Jigsaw transform. Proc. SPIE
**2013**, 8785, 87851Q. [Google Scholar]

**Figure 1.**(

**a**) Graphical effect of the Jigsaw transform (JT). (

**b**) Input image of the JT with a resolution of 256 × 256 pixels in grayscale. (

**c**) Input image after the first 16 × 16 Jigsaw transforms.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vilardy O., J.M.; Barba J., L.; Torres M., C.O. Image Encryption and Decryption Systems Using the Jigsaw Transform and the Iterative Finite Field Cosine Transform. *Photonics* **2019**, *6*, 121.
https://doi.org/10.3390/photonics6040121

**AMA Style**

Vilardy O. JM, Barba J. L, Torres M. CO. Image Encryption and Decryption Systems Using the Jigsaw Transform and the Iterative Finite Field Cosine Transform. *Photonics*. 2019; 6(4):121.
https://doi.org/10.3390/photonics6040121

**Chicago/Turabian Style**

Vilardy O., Juan M., Leiner Barba J., and Cesar O. Torres M. 2019. "Image Encryption and Decryption Systems Using the Jigsaw Transform and the Iterative Finite Field Cosine Transform" *Photonics* 6, no. 4: 121.
https://doi.org/10.3390/photonics6040121