Elsevier

Optics and Lasers in Engineering

Volume 62, November 2014, Pages 139-152
Optics and Lasers in Engineering

Asymmetric multiple-image encryption based on coupled logistic maps in fractional Fourier transform domain

https://doi.org/10.1016/j.optlaseng.2014.06.003Get rights and content

Highlights

  • ā€¢

    The phase retrieve process combined with coupled logistic map has high convergent speed.

  • ā€¢

    The PSNRs of all decrypted images can achieve high values to assure visual quality.

  • ā€¢

    The encryption scheme is based on asymmetric technique and the security is enhanced.

  • ā€¢

    The histograms of ciphertext and phase are nearly uniform and the security is enhanced.

Abstract

A multiple-image encryption scheme is proposed based on the asymmetric technique, in which the encryption keys are not identical to the decryption ones. First, each plain image is scrambled based on a sequence of chaotic pairs generated with a system of two symmetrically coupled identical logistic maps. Then, the phase-only function of each scrambled image is retrieved with an iterative phase retrieval process in the fractional Fourier transform domain. Second, all phase-only functions are modulated into an interim, which is encrypted into the ciphertext with stationary white noise distribution by using the fractional Fourier transform and chaotic diffusion. In the encryption process, three random phase functions are used as encryption keys to retrieve the phase-only functions of plain images. Simultaneously, three decryption keys are generated in the encryption process, which make the proposed encryption scheme has high security against various attacks, such as chosen plaintext attack. The peak signal-to-noise is used to evaluate the quality of the decrypted image, which shows that the encryption capacity of the proposed scheme is enhanced considerably. Numerical simulations demonstrate the validity and efficiency of the proposed method.

Introduction

With the rapid popularity of computer and internet, the exchange of information plays an important role in modern society. Images as an effective carrier of information have been widely used in various fields. The acquisition, transmission and processing of image have been seen at every corner of the digital age, and so image security issues have become increasingly serious and aroused a lot of attention. Since Refregier and Javidi proposed the optical image encryption based on input plane and Fourier plane random encoding [1], lots of schemes on optical image encryption have been put forward in other domains such as fractional Fourier transform (FrFT) [2], [3], [4], [5], [6], [7], [8], gyrator transform (GT) [9], [10], [11], [12], Fresnel transform (FrT) [13], [14], [15], and fractional Mellin transform [16], [17], [18]. Alfalou and Brosseau [19] analyzed the performance on different methods and pointed out many schemes can be used for compression simultaneously. Though most optical schemes have excellent properties such as parallel and multidimensional capability of signal processing, it should be pointed out that these schemes belong to the category of symmetric cryptosystems, where the keys are identical in the encryption and decryption processes. Due to the inherently linear property of mathematical or optical transformation, these schemes are vulnerable to the conventional attacks such as chosen plaintext attack. Additionally, most schemes mainly discuss the single image encryption, which reduce the efficiency when encrypting, storing and transmitting multiple images.

In order to relieve the network load, the double-image encryption has attracted lots of attentions. Li and Wang [20] proposed a double-image encryption based on iterative GT, where two plain images are encrypted into a single one as the amplitude of GT with different groups of angles simultaneously. Liu et al. [21], [22] suggested the double-image encryption schemes in the GT domain not only by using iterative random binary encoding but also by using random phase encoding and pixel exchanging. Additionally, Liu et al. [23] encrypted two plain images into the amplitude and phase of a complex function, respectively, in which the discrete fractional angular transform is used. Zhang and Xiao [24] designed a double optical image encryption by using the discrete Chirikov standard map which is utilized to scramble the pixel positions and intensity values, respectively. Li and Wang [25] proposed a double-image encryption based on discrete fractional random transform and chaotic maps, which can raise the efficiency when encrypting, storing or transmitting. Sui et al. [26] proposed a double-image encryption based on discrete fractional random transform, where a chaotic confusion-diffusion process is used to break the correlations between adjacent bit planes efficiently. Moreover, Wang and Zhao [27] suggested an asymmetric double-image encryption which has a high level of robustness against the specific attack.

With the development of double-image encryption techniques, more and more researchers pay their attentions to multiple-image encryption. Situ and Zhang [28], [29] proposed the multiple-image encryption schemes based on wavelength multiplexing and position multiplexing. Alfalou and Mansour [30] proposed a multiple images encryption scheme based on double random phase encoding, in which target images are multiplexed and encoded by using the iterative Fourier transform (FT). Subsequently, Alfalou and Brosseau [31] reported an algorithm to compress and encrypt multiple target images simultaneously based on a specific spectral multiplexing operation, where a fingerprint image is used as the first encryption key and a random phase key as the second key in order to achieve good security level. Similarly, Alfalou et al. [32] suggested a multiple-image encryption scheme based on the discrete cosine transform (DCT) and the specific spectral filtering technique, which implemented simultaneous fusion, compression and encryption of multiple images. Liu et al. [33] proposed an optical multi-image encryption based on frequency shift technique, where the lower frequency parts of the plain images are selected, shifted and encrypted by using double phase encoding in FrFT domain. Compared with other schemes, its optical implementation is efficient. Wang and Zhao [34] designed a multiple-image encryption based on the nonlinear phase truncation operations in FT domain, which can avoid the disadvantages of the classical double random phase encoding scheme and is vulnerable to conventional attacks such as chosen plaintext attack. Additionally, Wang and Zhao [35] proposed a fully phase multiple-image encryption based on superposition principle and digital holographic technique, where a real-valued plain image is encoded into a phase-only function (POF). Hwang et al. [36] proposed a multiple images encryption in FrT domain based on modified Gerchberg-Saxton algorithm (MGSA), which reduces the cross-talks of the decrypted images significantly. Based on MGSA, Chang et al. [37], [38] suggested the position multiplexing encryption schemes by using cascaded phase-only masks and Huang et al. [39] designed the scheme with architecture of two adjacent phase-only functions in FrT domain to increase capacity of the cryptosystem. Deng and Zhao [40] proposed a multiple-image encryption algorithm using phase retrieve algorithm and intermodulation in Fourier domain, which can avoid the cross-talk noise completely, but the convergent speed of iterative process should be further improved.

Recently, due to the excellent properties such as ergodicity, pseudo-randomness, sensitivity to initial conditions and control parameters, the chaotic maps are used to encrypt image in different transform domains, which can strengthen the nonlinearity of plain image in spatial and transform domains. Singh and Sinha [41], [42] proposed an optical image encryption schemes based on chaos not only in FrFT domain but also in GT domain. Liu and Wang [43] suggested a color image encryption based on spatial bit-level permutation, in which three channels of color image are confused and diffused by the high-dimensional chaotic map. Li et al. [44] designed a double-image encryption based on the chaos-based local pixel scrambling technique in GT domain, where two images are regards as the amplitude and phase of a complex function and then Arnold transform is used to scramble pixels at the local area. Wu et al. [45] proposed a four-image encryption method based on spectrum truncation, chaos and the multiple-order discrete fractional Fourier transform (MODFrFT), where the spectrum truncation is employed in discrete FT domain and the resultant performance is better than similar algorithm. Singh and Sinha [46] proposed a multiple images encryption based on chaos and multiple canonical transforms, where three linear canonical transforms such as FrFT, extended FrFT and FrT are utilized.

In this paper, an asymmetric multiple-image encryption scheme is proposed based on the coupled logistic maps in FrFT domain, in which the encryption keys are not identical to the decryption ones. First, a sequence of chaotic pairs is generated by using a system of two symmetrically coupled identical logistic maps and used to scramble the plain images. The POF of each scrambled image is retrieved by using an iterative process in the FrFT domain. Second, all POFs are modulated into an interim, which is transformed to the real-value ciphertext with stationary white noise distribution by using the FrFT and chaotic diffusion. Three random phase functions are used as encryption keys to retrieve the phase-only functions of plain images and three decryption keys are generated in the encryption process. Comprehensive application of the iterative process and chaos map makes the convergent speed faster when retrieving the POFs of plain images. Additionally, the cryptosystem enlarges the key space and achieves good encryption. Numerical simulations demonstrate the validity and efficiency of the proposed method.

The rest of this article is organized as follows. In Section 2, the basic principles and the processes of encryption and decryption are introduced in detail. In Section 3, numerical simulation results and security analysis are given. Finally, the conclusion is given in Section 4.

Section snippets

Logistic map and two-coupled logistic map

Chaos theory is a famous theory on the study of nonlinear dynamics, in which seemingly random events are actually predictable from simple deterministic equations. The dynamical systems are established based on various chaos functions such as logistic map, Lorenz attractors and so on. A chaos function has three properties: (1) it is sensitive to initial conditions; (2) it is topologically mixing; (3) its periodic orbits are dense. With a chaotic map, a large number of random iterative values

Numerical simulation and security analysis

The proposed multiple-image encryption is carried out to verify the feasibility of the cryptosystem with nine plain images shown in Fig. 2. Two groups of fractional orders are set as Ī±1=0.2, Ī±2=Ī±1+0.4, Ī²1=Ī±1+0.3, Ī²2=Ī±2+0.1 and Ī±3=Ī±1. The initial values of two-coupled logistic map are set as x0=0.21, y0=0.83 and K is set to 2000. The system parameters of two-coupled logistic map are set as p=3.56995 and Īµ=āˆ’0.471. The MSE threshold as the convergent criterion is set to 1.0eāˆ’9. The ciphertext

Conclusion

In summary, a multiple-image encryption scheme is proposed based on asymmetric technique, in which the decryption keys are not identical to the encryption ones. Three random phase functions are employed as encryption keys and used to retrieve the POFs of plain images based on the iterative phase retrieval process. All POFs are combined into a complex matrix, which is transformed to the real-value ciphertext with stationary white noise distribution. Meanwhile, three decryption keys are generated

Acknowledgments

This work was supported by Chinese NSFC under grant 61172123, Fok Ying Tung Education Fund under grant 141119 and the Foundation of Shaanxi Education Department of Shaanxi Province under grant 11JK1032 and 2010JK732.

References (56)

  • H. Li et al.

    Double-image encryption based on iterative gyrator transform

    Opt Commun

    (2008)
  • Z. Liu et al.

    Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains

    Opt Laser Technol

    (2013)
  • Z. Liu et al.

    Double image encryption by using Arnold transform and discrete fractional angular transform

    Opt Lasers Eng

    (2012)
  • Y. Zhang et al.

    Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform

    Opt Lasers Eng

    (2013)
  • H. Li et al.

    Double-image encryption based on discrete fractional random transform and chaotic maps

    Opt Lasers Eng

    (2011)
  • L. Sui et al.

    Double-image encryption using discrete fractional random transform and logistic maps

    Opt Lasers Eng

    (2014)
  • Z. Liu et al.

    Optical multi-image encryption based on frequency shift

    Optik

    (2011)
  • X. Wang et al.

    Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain

    Opt Commun

    (2011)
  • X. Wang et al.

    Fully phase multiple-image encryption based on superposition principle and the digital holographic technique

    Opt Commun

    (2012)
  • H.T. Chang et al.

    Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain

    Opt Commun

    (2011)
  • J.J. Huang et al.

    Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain

    Opt Laser Technol

    (2012)
  • X. Deng et al.

    Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain

    Opt Laser Technol

    (2012)
  • N. Singh et al.

    Optical image encryption using fractional Fourier transform and chaos

    Opt Lasers Eng

    (2008)
  • N. Singh et al.

    Gyrator transform-based optical image encryption, using chaos

    Opt Lasers Eng

    (2009)
  • H. Liu et al.

    Color image encryption using spatial bit-level permutation and high-dimension chaotic system

    Opt Commun

    (2011)
  • H. Li et al.

    Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform

    Opt Lasers Eng

    (2013)
  • J. Wu et al.

    Four-image encryption method based on spectrum truncation, chaos and the MODFrFT

    Opt Laser Technol

    (2013)
  • N. Singh et al.

    Chaos based multiple image encryption using multiple canonical transforms

    Opt Laser Technol

    (2010)
  • Cited by (57)

    • EMOTE ā€“ Multilayered encryption system for protecting medical images based on binary curve

      2022, Journal of King Saud University - Computer and Information Sciences
    • Optical encryption scheme for multiple-image based on spatially angular multiplexing and computer generated hologram

      2020, Optics and Lasers in Engineering
      Citation Excerpt :

      Current multiple-image encryption technology is mainly based on the following methods or principles: Multiplexing, Digital Holography, Compressed Sensing, Chaos and special optical transformation. For example, Situ introduced wavelength multiplexing to achieve multiple-image encryption [11], Sui proposed a multiple-image encryption algorithm based on phase recovery technology and phase mask multiplexing [12]. Xu proposed a multiple-image encryption method based on random amplitude plate and Fresnel hologram [13], Wan proposed a multiple-image compression holographic algorithm based on improved Mach-Zehnder interferometer [14].

    • An overview of encryption algorithms in color images

      2019, Signal Processing
      Citation Excerpt :

      Step 3: Repeating Step 2 until reaching cipher-image. The conventional chaotic systems utilized in image encryption processes are Lorenz map [81,82], Baker map [83,84], Arnold's cat map [85,86], HĆ©non map [87,88], Logistic map [51,89], Chee-Lee system [90], Hyper-chaotic system [80,91], Quantum Logistic map [92,93], Multiple coupled map lattices [94], Tent and sine map [95], etc. In the following, some methods proposed for color image encryption using chaotic system will be reviewed.

    View all citing articles on Scopus
    View full text