## Abstract

Recently, a number of double-image cryptosystems have been developed. However, there are notable security performance differences between the two encryption channels in these algorithms. This weakness downgrades the security level and practicability of these cryptosystems, as the cryptosystems cannot guarantee all the input images be transmitted in the channel with higher security level. In this paper, we propose a novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains. The two input images are firstly shuffled by the proposed cross-image pixel scrambling approach, which can well balance the pixel distribution across the input images. The two scrambled images will be encoded into the real and imaginary parts of a complex function, and then converted into gyrator domains. An iterative architecture is designed to enhance the security level of the cryptosystem, and the cross-image pixel scrambling operation is performed to the real and imaginary parts of the generated complex encrypted data in each round. Numerical simulation results prove that a satisfactory and balanced security performance can be achieved in both channels.

© 2014 Optical Society of America

## 1. Introduction

With the dramatic development of communication technologies, image sharing and exchange across Internet have become much more prevalent than the past. Cryptographic approaches are therefore critical for secure image transmission and storage over public networks. During the past decades, optical systems are of growing attraction for image encryption due to their high speed and parallel processing advantages. In 1995, Refregier and Javidi proposed the double random phase encoding (DRPE) architecture based on the 4*f* optical system to encrypt the primary image into stationary white noise [1]. In [2], Unnikrishnan and Singh introduced the DRPE structure into fractional Fourier domain, and then the fractional Fourier transform (FrFT) has shown its advantages in the optical security areas and a number of algorithms for image encryption are subsequently proposed [3–9]. Besides, researchers have developed optical image encryption schemes in gyrator transform domains [10–13], especially after Liu Z et al. addressed the numerical simulation difficulties of gyrator transform in [14]. Some other strategies, such as digital holography [15–17], interference [18–20], diffractive imaging [21], image sharing [22, 23] and watermarking [24, 25] techniques are also employed to build secure image encryption schemes. To satisfy the requirements of transmitting multiple images simultaneously, multiple-image encryption algorithms have been researched in recent years. In 2005, Situ and Zhang firstly proposed a multiple-image encryption scheme using wavelength multiplexing [26], whereas the subsequent multiple-image encryption schemes are mostly based on amplitude and phase encoding of the original images, such as the double-image encryption schemes in [27–32].

From the standpoints of the sender and the receiver, the block diagram of a typical secure double-image communication system can be simplified illustrated by Fig. 1. All the encryption, decryption and the communication link procedures (with various kinds of attacks) can be simplified observed as two secure image transmission channels, denoted as channel 1 and channel 2, respectively. For image cryptosystems, the transmission performances not only include the traditional communication factor, such as noise resistance, but also the cryptography indices, such as key sensitivity and tolerance against the occlusion attacks. In the following, we use security performance to represent both the traditional communication factor and security performance of a cryptosystem. It is important to note that, when evaluating the security performance of a double-image cryptosystem, we must make an overall consideration of the two channels, so as to estimate the performance systematically and scientifically. That means, if the security performances in the two channels are not the same, the lower security performance servers as the actual security performance of the whole cryptosystem. That’s because the cryptosystem cannot guarantee that all the input images be transmitted in the channel with higher security level. As to the double-image cryptosystems in [27–32], there are obvious security performance differences between the two secure transmission channels. Taking Tao’s scheme in [27] as an example, the security performance in the phase-based channel is much better than that in the amplitude-based channel. And hence, the researchers pointed out that, “an image with higher security requirement is suggested to act as the phase-based image when using the proposed encryption method due to these properties” [27]. However, in practical circumstances, it is improbable to ask the senders to distinguish what images are with higher security requirements and what images can be transmitted at lower security level. And hence, the difference of the security performances between the two channels downgrades the overall security level and the practicability of those double-image cryptosystems. In this regard, if we can design a double-image cryptosystem that can well balance the security performances in the two channels, the security level and the practicability of the whole cryptosystem will be promoted accordingly.

In this paper, we propose a novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains. The two input images will be firstly scrambled using the proposed cross-image pixel scrambling approach. By using this strategy, the two images are treated as a whole, and the pixels in the original images will be shuffled across each other. In other words, each of the two plain images will be divided into two fragmental parts, and will be transmitted in the two channels together with the fragmental parts of the other original image. Besides, simulation results prove that the pixel distributions of the scrambled images are roughly the same, which reflects that the information in the two original images has been balanced scattered into the scrambled images. Therefore, when confronted with various attacks or secret key mismatching, the security performance differences between the channels can only affect the scrambled images, and will also be balanced scattered into the two recovered images after the inverse cross-image pixel scrambling procedure, as shown in Fig. 2. In our cryptosystem, the two scrambled images will be encoded into the real and imaginary parts of a complex function, and then converted into gyrator transform domains. An iterative architecture is designed for enhancing the security level of the cryptosystem, and the cross-image pixel scrambling operation will be performed to the real and imaginary parts of the produced complex encrypted data in each iterative round. The cryptosystem can be implemented with an electro-optical hybrid setup, in which the pixel scrambling is performed with the help of a computer, whereas the gyrator transform can be achieved by using an optimized flexible optical system which contains only three generalized lenses with fixed distance between them [33]. Numerical simulations have been performed, the results demonstrate that the proposed scheme has higher security level and can well address the weakness of the security performance differences between the transmission channels in traditional optical double-image cryptosystems.

The remaining of the paper is organized as follows. In next section, the proposed double-image encryption scheme will be given out in detail. Numerical simulation results and discussions are reported in Section 3. Finally, conclusions will be drawn in the last section.

## 2. The proposed double-image encryption scheme

#### 2.1 Cross-image pixel scrambling strategy

In this section, cross-image pixel scrambling approach, which can shuffle the pixels among the original images, is proposed. The main effect of this strategy is to scatter the pixels of the original images into each other, and hence the information stored in the two images will be balanced distributed into the scrambled images.

Suppose the two original images are with size of *M* × *N*, denoted as *img1* and *img2*, and the coordinates of rows and columns range from 1 to *M* and 1 to *N*, respectively. The detailed procedures of cross-image pixel scrambling approach are described as follows.

(1) The original images are firstly combined into a new image, *img3*, with size of *M* × 2*N*, where *img1* lying on the left whereas *img2* is placed on the right side. As shown in Fig. 3, Figs. 3(a) and 3(b) with 512 × 360 pixels serve as *img1* and *img2*, whereas Fig. 3(c) is the combined image with the size of 512 × 720.

(2) Generate the row confusion vector for *img3*, denoted as *R* = (*r*_{1}, *r*_{2},…, *r _{M}*), and the value of

*r*(1≦

_{m}*m*≦

*M*) is restricted within the region [

*m*,

*M*], and can be calculated by

*floor*(

*x*) returns the value nearest integers less than or equal to

*x*, mod(

*x*,

*y*) returns the remainder after division,

*state_variable*is the current state of a chaotic map. In our scheme, the applied chaotic state variables are produced from Chebyshev map, as described bywhere

*k*and

*x*are the control parameter and state value, respectively. If one chooses $k\in [2,\infty ),$ the system is chaotic. The initial value

_{n}*x*

_{0}and the control parameter

*k*serve as the secret key of Chebyshev map.

(3) Rearrange each row of *img3* from the top to the bottom, according to confusion vector *R*, that means move the first row to *r*_{1}th row, …, move the last row to the *r _{M}*th row.

(4) Rearrange each column of the image produced after (3) in the same way, where all the values *M* vary to 2*N*.

(5) Divide the scrambled image produced in (4) into two images, with the left half area treated as the scrambled *img1* whereas the right half part is the scrambled *img2*.

The scrambled version of the combined image is shown in Fig. 4(a), and the divided images, scrambled *img1* and scrambled *img2*, are demonstrated in Figs. 4(b) and 4(c).

The histograms of the original images and the scrambled images are shown in Fig. 5, with the purpose to clearly demonstrate the information balancing property of the cross-image pixel scrambling strategy. The information balancing property of the cross-image pixel scrambling strategy can be proved in two aspects. (1) The histograms of the original images and the corresponding scrambled images are different. For example, the histogram of the original Barb image, shown in Fig. 5(a), shows no similarity with that of the scrambled image, depicted in Fig. 5(c). That means the information distributions in the original images have been significantly changed after the cross-image pixel scrambling operation. (2) The histograms of the two scrambled images are roughly the same with each other, as shown in Figs. 5(c) and 5(d), which means the pixels of the two original images have been balanced distributed into the two scrambled images. In other words, after the cross-image scrambling procedure, the two distinct original images have been converted into two roughly identical images in terms of statistical properties.

#### 2.2 Gyrator transform

Gyrator transform (GT) is mathematically defined as a linear canonical transform which produces a rotation in the twisting (position – spatial frequency) phase planes [10, 33]. The gyrator transform at parameter *α*, which is called as the rotation angle, of a two dimensional function *f* (*x*, *y*) is calculated as

The function *F* (*u*, *v*) is the output of the transform. The GT has some properties similar to fractional Fourier transform, and is additive and periodic with respect to the angle *α*. The transform can be achieved by using an optimized flexible optical system which contains only three generalized lenses with fixed distance between them, and the angle *α* is changed by rotation of the cylindrical lenses which form the generalized lenses [10].The GT can also be computer simulated by using phase-only filtering, Fourier transform and inverse Fourier transform [14]. The inverse GT corresponds to the GT at angle -*α*.

#### 2.3 The proposed double-image encryption scheme

The flowchart of the proposed double-image encryption algorithm is given out in Fig. 6. Let *img1* and *img2* represent the two original normalized images to be encrypted together, the procedures of the encryption process are described as follows.

**Step 1**: Shuffle the two images using the cross-image pixel scrambling approach, the corresponding scrambled images are denoted as *I*_{1} and *I*_{2}, respectively.

**Step 2**: Encode the two scrambled images into a complex function, defined as

**Step 3**: Convert the complex function using the gyrator transform at angle *α*, and the resultant complex data is expressed as

**Step 4**: Repeat step 2 and step 3 *N* times to satisfy the security requirements, using *A*_{1} and *A*_{2} as the input data in each iterative round. Finally, the encrypted data *E _{n}* is produced.

A possible optoelectronic hybrid system adopted to implement the proposed method is shown in Fig. 7, in which the cross-image pixel scrambling operation will be performed with the help of a computer and the other processes can be optically implemented. In the encryption procedure, the two original images are firstly cross-image scrambled with the help of a computer to produce the combined complex-valued input signal. The combined signal will be then imported into spatial light modulators (SLM), and be illuminated by collimated light from the beam expander (BE) located behind the laser. The input complex data is then received by the GT module. Rodrigo’s achievements in [10, 33] are employed to perform the GT optically, using three generalized lenses. Each of the generalized lenses is a combination of two convergent thin cylindrical lenses with the same power. The transformation angle *α* is controlled by appropriate rotation of these two thin cylindrical lenses. The first and third generalized lenses are identical and denoted by *L*_{1}, with the focal length *f*_{1} equal to the distance *z* between two consecutive generalized lenses (the first and second or the second and third) of the optical implementation. The second generalized lens *L*_{2} has a focal length *f*_{2} = *z*/2 [34]. The resultant data is recorded by in-line holography, and then sent back into the computer to complete the iterative encryption operations.

In this scheme, the initial value *x*_{0} and the control parameter *k* of Chebyshev map, and the rotation angle *α* of the gyrator transform serve as the secret keys.

## 3. Numerical simulation and discussion

The testing images used for simulation are Barb and Boats images with size of 360 × 512, corresponding to the input images, *img1* and *img2*, shown in Fig. 8. The total iterative count of the encryption is fixed at 3 rounds, and the rotation angle is taken at 0.51. The initial value *x*_{0} and the control parameter *k* of the Chebyshev map are randomly selected as 0.111 and 5.0, respectively. Our simulation platform is a personal computer with an Intel(R) Core(TM) i5 CPU (2.27GHZ), 2GB memory and 320GB hard-disk capacity, using Matlab R2010a.

The amplitude of the complex-valued encrypted image is demonstrated in Fig. 8(c), which are randomly noise-like images. Figures 8(d) and 8(e) illustrate the decrypted images using the correct keys, with no noise or distortion occurs in the encrypted data transmission. They are consistent with the original images.

Next, we will analyze the security performances of the proposed scheme in various aspects. To express the performance indices numerically, we firstly introduce the mean square error (MSE) function [31], as described in Eq. (6), where *I _{o}*(

*m*,

*n*) and

*I*(

_{d}*m*,

*n*) are the normalized original and recovered images at coordinate (

*m*,

*n*), respectively,

*M*and

*N*represent the width and length of the images.

*I*and the decrypted image

_{o}*I*, when the encrypt image undergoes various attacks or key mismatching. Besides, we simply use channel 1 and channel 2 to represent the two secure transmission channels of our cryptosystem, corresponding to the transmission channels for

_{d}*img1*and

*img2*, respectively.

#### 3.1 Robustness to noise perturbation

The robustness against noise perturbation of the proposed scheme is researched in this subsection. Figures 9(c) and 9(d) are the decrypted images retrieved from the encrypted data distorted by zero-mean white additive Gaussian noise with a standard deviation of 0.01. The MSEs between the original and the decrypted images of Barb and Boats are both 0.005. The decrypted images when the ciphered image is perturbed by the Gaussian noise with a standard deviation of 0.05 are shown in Figs. 9(e) and 9(f), and the corresponding MSEs are 0.0250 and 0.0251, respectively.

Figure 10 illustrates the curves of the MSE between the corresponding original and decrypted images versus the noise standard deviation. The step size of noise standard deviation is 0.01, and the deviations vary from 0 to 0.5. Obviously, the MSE curves of the two images (channels) overlap almost completely, which means the robustness to the noise perturbation of the two channels is roughly the same with each other. This property can be viewed as a significant improvement of the traditional double-image encryption schemes.

#### 3.2 Tolerance against occlusion attacks

In this section, we check the tolerance against the loss of encrypted data. Figures 11 and 12 demonstrate the recovered images when 25% and 50% of the encrypted data are occluded, respectively. When 25% of the encrypted data is blocked, the original images can be recovered with the MSEs of 0.0568 and 0.0546 for Barb and Boats, respectively.

When 50% of the encrypted data is occluded, the retrieved images, shown in Figs. 12(b) and 12(c), are still recognizable and the corresponding MSEs between the recovered and the original images are 0.1145 and 0.1073, respectively. As the encryption scheme distributes the original images over the entire encrypted image, and hence it can provide robustness against occlusion attacks. Besides, with the help of the cross-image pixel scrambling technique, the MSEs of the recovered images are roughly the same, and hence the tolerances to the occlusion attacks of the transmission channels are well balanced.

#### 3.3 Sensitivity of the rotation angle

In this section, we study the sensitivity of the rotation angle of the gyrator transform, which is part of the secret key. By applying the correct parameters of Chebyshev map to no distorted encrypted image, the MSEs when using different decryption angles are calculated and depicted in Fig. 13. Besides, the recovered images when the decrypted operation are performed using angle *α* = −0.52, which is slightly different from the correct value −0.51, are demonstrated in Fig. 14, as an example. The results indicate that the two channels are both highly sensitive to the rotation angle.

Note that, there is tiny MSE difference between the two channels, as depicted in Fig. 13. However, the difference in our scheme is much slighter compared with that in the Tao’s scheme [27]. Numerically, the difference ratio in our scheme is about 5%, whereas the difference ratio is approximately 40% in [27]. Besides, the sensitivity difference in our scheme is not fixed to the channels, which means channel 1 is not always more sensitive to the rotation angle than channel 2. The sensitivity difference of the rotation angle in our cryptosystem is randomly related with the original images. For example, when using Boats as *img1* and Barb as *img2*, the MSE curves with various values of the rotation angle are shown in Fig. 15. The sensitivity of Barb image is also better than Boats, even though it is encrypted and transmitted in channel 2 at this time. This property can also prove that the key sensitivity performances of the two channels have been well balanced, as there is no channel which always possesses a better security performance than another one.

#### 3.4 Sensitivity of the Chebyshev map

Here, the contribution of the Chebyshev map, the initial value *x*_{0} and the control parameter *k,* on the security of the proposed cryptosystem is researched. The sensitivity of the initial value *x*_{0} can be observed from Fig. 16, which shows the decrypted images with a slightly different initial value, 0.111 + 10^{−15}. The MSEs of between the decrypted and the original images are 0.4571and 0.4281 for Barb and Boats, respectively. On the other hand, the decrypted Barb and Boats images by changing the value of *k* from 5.0 to 5.0 + 10^{−15} are shown in Fig. 17, and the corresponding MSEs are 0.4560 and 0.4293, respectively. The results prove that, the cryptosystem is high sensitive to the parameters of the Chebyshev map. Even a slightly mismatching in the parameters can result in totally incorrect and noise-like decrypted images.

For Chebyshev map, *x*_{0} can be any value between −1 and 1. According to the IEEE floating-point standard [35], the computational precision of the 64-bit double-precision number is about 10^{−15}, so the total number of possible *x*_{0} values is about 2 × 10^{15}. Though *k* can be any real value greater than 2.0 in theory, the range of *k* should be restricted to 2π to prevent Chebyshev map from producing periodic orbits [36], so the total number of the possible valued of *k* is approximately 2π × 10^{15}. Therefore, the total number of the possible secret key combination is approximately 4π × 10^{30} for Chebyshev map, which is large enough to resist brute-force attack.

#### 3.5 Further discussion

Generally, there are two procedures in conventional double image cryptosystems. The first one is an image encoding procedure, which is generally used to combine the original images into a complex function. The second procedure is a kind of optical transform module, with the purpose to convert the combined complex function into various transform domains. An effective image encoding algorithm plays the core role for balancing the secure performances in the two channels. The proposed cross-image pixel scrambling is an approach of this type. In the present paper, we use this strategy collaborated with the gyrator transform, and the simulation results prove the effectiveness of the cross-image pixel scrambling technique. Besides, researchers have also employed this method to work with fractional Fourier transform, and the simulation results are also very satisfactory. Interested readers can try to apply the cross-image pixel scrambling strategy as a pre-processing layer, and work with some other double-image encryption techniques, to investigate the effectiveness and universality of the novel approach.

## 4. Conclusions

In this paper, we present a novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains. Cross-image pixel scrambling approach is firstly proposed, with the purpose to redistribute the pixels among the original images. By using this strategy, the two images are treated as a whole, and the pixels in the original images are shuffled into each other. The two confused images are encoded into the real part and the imaginary part of a complex function and then converted into gyrator domains. An iterative architecture is designed to enhance the security level, and the cross-image pixel scrambling operation is performed to the real and imaginary parts of the produced complex encrypted image in each iterative round. Numerical simulations have proved that the proposed scheme can well address the weakness of the security difference between the two channels in traditional double-image cryptosystems.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 61271350, 61374178, 61202085), the Fundamental Research Funds for the Central Universities (No. N120504005), the Liaoning Provincial Natural Science Foundation of China(No. 201202076), the Specialized Research Fund for the Doctoral Program of Higher Education(No. 20120042120010) and the Ph.D. Start-up Foundation of Liaoning Province, China (Nos.20111001, 20121001, 20121002).

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