## Abstract

We present a novel image hiding method based on phase retrieval algorithm under the framework of nonlinear double random phase encoding in fractional Fourier domain. Two phase-only masks (POMs) are efficiently determined by using the phase retrieval algorithm, in which two cascaded phase-truncated fractional Fourier transforms (FrFTs) are involved. No undesired information disclosure, post-processing of the POMs or digital inverse computation appears in our proposed method. In order to achieve the reduction in key transmission, a modified image hiding method based on the modified phase retrieval algorithm and logistic map is further proposed in this paper, in which the fractional orders and the parameters with respect to the logistic map are regarded as encryption keys. Numerical results have demonstrated the feasibility and effectiveness of the proposed algorithms.

© 2014 Optical Society of America

## 1. Introduction

The double random phase encoding (DRPE) in Fourier domain proposed by Refregier and Javidi in 1995 is one of the most widely used optical encryption technique [1]. It has already been extended from the Fourier domain into other domains over the past two decades [2, 3], such as fractional Fourier domain [4], Fresnel domains [5–7] and gyrator transform [8]. The indispensable holography techniques, however, increase the complexity of the encryption systems. More importantly, investigations have shown that the DRPE scheme and many of its variations are vulnerable to some attacks for their linearity and symmetry [9, 10]. Thus, improving the security without significantly increasing the complexity of the cryptosystems has become a very important research goal [11–14]. Recently, the nonlinear DRPE methods based phase-truncation have been proposed to enhance the security of DRPE [15–18]. But the optical setups are still complex and there are still security concerns [19, 20]. From the point view of information sharing and data transmission, information of the primary image to be encoded has in fact been hidden into one amplitude mask (AM) and two phase-only masks (POMs) in those linear and nonlinear DRPE-based methods [4–6, 8, 11–15].

In order to reduce the complexity of system configuration and the amount of data being transferred, image encryption methods based on phase retrieval algorithm [21–28], interference principle [29] and information pre-choosing [30] have been proposed in recent years, in which the original image can be digitally encrypted into several POMs. The recovery of the secret image can be implemented optically or digitally. Before optical implementation, one needs to produce POMs by using micro-optics fabrication techniques [31]. Unfortunately, undesired information disclosure has been discovered to reduce the security performance of those systems when we shift our focus from preventing the keys from being revealed in different kinds of attacks to the keys themselves. The information disclosure in some of the masks may make the other keys or the ciphertext useless. Indeed, ensuring that no valuable information about the secret image could be observed unless all the POMs and AMs are correctly placed in the verification system is not an easy task, especially for most of phase-truncated-based encryption methods where the risk of information disclosure is hidden in the case of one-image encoding [32]. For interference-based image hiding methods, post-processing of the POMs and digital inverse computation after optical decryption are usually used as improvement measurements [33–37]. But for the phase-truncation-based encryption methods, it can be proved that the problem of information disclosure can’t be thoroughly solved by using imaginary-part truncation [17] and steganographic approach based on iterative phase retrieval [28]. Thus, it makes sense to cast about for some other way to calculate the AMs and POMs without the problem of information disclosure. The nonlinear phase-truncated based DRPE, however, seems to be put off in the course of this casting about for its serious information disclosure as we remarked above. But it should be mentioned that the problem of information disclosure does not imply that the phase-truncated cryptosystem does not work. It means that a new process to eliminate this risk must be generated [32]. In practice, we would prefer using it to merely rejecting it.

In this paper, we first propose an image hiding method based on a phase retrieval algorithm under the framework of nonlinear DRPE in fractional Fourier domain, in which two cascaded phase-truncated fractional Fourier transforms (FrFTs) are used for each iteration loop. Two POMs can be efficiently determined in this method and no undesired information can be observed from any one of them. It should be mentioned that there is no need for phase recording in this cryptosystem since phase truncation only appears in the digital nonlinear encryption process. No post-processing of the POMs is required in encryption process and no digital inverse computation is needed in the decryption in our method. The decryption process is designed to be linear so that it can be implemented optically based on linear DRPE in fractional Fourier domain. In this paper, we also present an effective, safe and practical approach to reduce the load of key transmission by using logistic map and modifying the phase retrieval algorithm. In this approach, the POM obtained in fractional Fourier domain can be treated as encrypted result while the fractional orders and the two parameters with respect to logistic map are kept as encryption keys. Two different methods are presented in Sections 2 & 3, respectively.

## 2. Image hiding based on phase retrieval algorithm under the framework of nonlinear DRPE in fractional Fourier domain

Let’s briefly introduce the FrFT first. The one-dimensional FrFT of $\alpha $ order of a function $f(x)$ is defined as [38]

*x*-axis. The extension of the definition to two-dimensional signals is straightforward and simple that can be expressed as

Since the image hiding method is based on phase retrieval algorithm under the framework of nonlinear DRPE in fractional Fourier domain, it is necessary to give a brief introduction to the nonlinear DRPE. The flowchart of the encryption process of the nonlinear DRPE in fractional Fourier domain is shown in Fig. 1, where $f(x,y)$ represents the amplitude distribution of the secret image and the operators $\otimes $, PT and PR denote multiplication, phase truncation and phase reservation, respectively. Here phase truncation means retaining the amplitude part of a complex function but truncating its phase part. Phase reservation on a complex function, in turn, means retaining the phase part of the function but getting rid of its amplitude part. Two computer-generated random phase masks (RPMs) $R$ and ${R}^{\prime}$ are used as two encryption keys.

As shown as in Fig. 1(a), the amplitude distribution denoted by $g(u,\nu )$ and the encrypted result $E(x,y)$ can be respectively written as

Two phase keys produced in the encryption process, i.e., $P(u,\nu )$ and ${P}^{\prime}(x,y)$ are given by

The decryption process as shown in Fig. 1(b) can be described by the following two steps:

In the following, we describe the image hiding method based on phase retrieval algorithm under the framework of the nonlinear DRPE above. The flowchart of the $k\text{th}\left(k=1\text{,2,3}\cdots \right)$ loop of the iterative process for image hiding is illustrated in Fig. 2, where ${f}^{\prime}(x,y)$ represents the amplitude distribution of the decrypted image. In the first iteration stage, the encryption keys are two RPMs denoted by ${R}_{1}$ and ${{R}^{\prime}}_{1}$, which will be incessantly updated in the following iterations.

As shown in Fig. 2, every phase key is changeable and will be updated after each iteration. The process for the $k\text{th}$ iteration can be summarized as follows:

- (1) Perform FrFT with the fractional order $\alpha $ to the product of the amplitude of the input image $f(x,y)$ and the first POM ${R}_{k}$. The amplitude and phase distributions of the resultant complex function can be respectively written as
- (2) Perform FrFT with the fractional order $\beta $ to the product of the amplitude ${g}_{k}(u,\nu )$ and the second POM ${{R}^{\prime}}_{k}$. The phase distribution obtained by $\text{PR}$ operation is given by
- (4) Likewise, the amplitude and phase distributions of the complex amplitude resulting from performing FrFT with the fractional order $-\alpha $ to the product of ${{g}^{\prime}}_{k}(u,\nu )$ and the POM ${P}_{k}(u,\nu )$ given by Eq. (12) are respectively written as

By assumption the iterative computation stops right after the $m\text{th}$ iteration loop. By this we can obtain the final decrypted amplitude image ${{f}^{\prime}}_{m}$ and four POMs, ${R}_{m+1}$, ${{R}^{\prime}}_{m+1}$, ${P}_{m}$ and ${{P}^{\prime}}_{m}$. As indicated by Eq. (15) and Eq. (16), the two POMs obtained by phase-truncation, ${P}_{m}$ and ${{P}^{\prime}}_{m}$, can be used as two decryption keys to obtain ${{f}^{\prime}}_{m}$. However, the optical setup for a nonlinear decryption process is too complex [15]. Moreover, it can be seen from Fig. 2 that the phase key, ${P}_{m}$, is just the phase part of the spectrum. Thus, we still have to consider the problem of undesired information disclosure since it is emerging as an issue across many of the nonlinear DRPE-based encryption methods [32]. Two phase keys for a linear decryption without the problem of information disclosure are what we really want. In fact, it's not difficult to do that. Let us define three functions designated ${f}^{\prime}(x,y)$, ${K}_{1}(u,\nu )$ and ${K}_{2}(x,y)$ by the following equations:

With the help of Eqs. (14)–(20), thus, we have

## 3. Image hiding using logistic map and phase retrieval algorithm

As can be shown in Fig. 3, the POM ${K}_{2}(x,y)$ and the two fractional orders can be regarded as decryption keys when the POM ${K}_{1}(u,\nu )$ is kept as the encrypted result. In the following, we will try to reduce the load of data transmission based on logistic map and modification of the above approach. The logistic map that we adopt in this paper is given in the iterative form by

where $\mu \in [0,4]$ is the bifurcation parameter and ${x}_{n}\in [0,1]$ is the sequence value. When $3.5699456\le \mu \le 4$, slight variations in the initial value yield dramatically different results over time [39]. A two-dimensional chaotic random phase mask (CRPM) can be generated by using a logistic map. We first use the chaos function to generate a non-converging sequence with the initial value ${x}_{0}$ and the bifurcation parameter $\mu $ and then take column-wise from the sequence to obtain a matrix $S(x,y)$. Subsequently, the computer-generated CRPM can be represented by $C(x,y)=\mathrm{exp}[i2\pi S(x,y)]$. Obviously, the bifurcation parameter $\mu $, and the sequence value ${x}_{0}$, are essential for the correct retrieval of the CRPM.Suppose the phase key ${K}_{2}(x,y)$ is replaced by a computer-generated CRPM given above. By this it means that the two parameters corresponding to the CRPM and the fractional orders now form the key size for decryption, which inexplicitly implies that the CRPM would be fixed in the iterations once it has been produced. Therefore, the phase retrieval algorithm shown in Fig. 2 may need to be correspondingly adjusted, i.g., ${{R}^{\prime}}_{k}$ and the FrFT of order $\beta $ should be removed. Figure 4 shows the flowchart of the $k\text{th}\left(k=1\text{,2,3}\cdots \right)$ iteration after the adjustments.

In the $k\text{th}$ iteration, the definitions of the functions appeared in Fig. 3 should now be rewritten as

Note that the POM ${P}_{k}(u,\nu )$ still can be represented by Eq. (6). Likewise, after $n$ iterations, we have the decrypted amplitude image ${{f}^{\prime}}_{n}$ and three POMs, ${R}_{n+1}$, ${{R}^{\prime}}_{n+1}$, ${P}_{n}$. In much the same way as in the preceding section, the decrypted image is defined as ${f}^{\prime}(x,y)={{f}^{\prime}}_{n}(x,y)$ and the encrypted result is constructed as $E(u,\nu )={{R}^{\prime}}_{n+1}^{\ast}(u,\nu ){P}_{n}(u,\nu )$. In this new method, similar reasoning leads to the expression

## 4. Numerical simulations and discussion

Numerical simulations are performed to test the feasibility of the proposed algorithms under the situation of limited resources. A normalized grayscale image, Lena, as shown in Fig. 6(a) with a size of $256\times 256$, is used as the secret image. For simplicity, we choose $\alpha =\beta $ in this section. The two POMs generated from Lena using the first method, ${K}_{1}(u,\nu )$ and ${K}_{2}(x,y)$, are shown in Fig. 6(b) and Fig. 6(c), respectively. The corresponding fractional orders are $\alpha =\beta =0.6$ and the total iteration number is 50. The target image can be reconstructed based on these two POMs via Eq. (21).

The mean square error (MSE) and the correlation coefficient (CC) are applied to evaluate the similarity between the decrypted image ${f}^{\prime}(x,y)$ and its original image $f(x,y)$, which are respectively defined by

Now we check for the problem of undesired information disclosure. When only the POM ${K}_{1}(u,\nu )$ is placed in the verification system, the decrypted amplitude image produced by performing FrFT to ${K}_{1}(u,\nu )$ with the correct fractional orders $-\alpha =-0.6$ is shown in Fig. 9(a).Figure 9(b) demonstrates the decrypted result when only the POM ${K}_{2}(x,y)$ is placed in the verification system. Fortunately, no valuable information can be observed from both of them. Figure 10 shows the blind decryption images by using an arbitrarily selected phase key. When one of the two POMs is replaced by an incorrect random phase key during image decryption, the decryption process unavoidably ends in failure.

The influence of deviation in the fractional order from their right value has been further studied. Figure 11 shows the MSE curve with respect to different fractional orders. It can be found that the MSE value approximates to zero when the fractional orders approach to the correct one, i.e., $-\alpha =-0.6$. If there is a deviation from the correct order, the MSE increases rapidly. But it should be mentioned that if the two POMs are moving closer and closer to their right places, one can imagine that this might lead to a series of blurred images in the output plane. We investigate this issue and find that when the fractional order used for decryption is very close to the correct one, a serious blurred and indecipherable silhouette can be obtained as expected in the decryption process, which can be shown in Fig. 11(b). But the decrypted result as shown in Fig. 11(c) will be unrecognizable if the deviation of fractional order is a little bit larger than that. Actually, one can set $\alpha \ne \beta $ in the encryption process for increasing the difficulty in decryption.

Since only two POMs are produced in our method, it is impossible for an attacker to obtain one POM by using another POM based on amplitude-phase retrieval algorithm, where two constraints are at least required [19]. An example of a brute attack based on amplitude-phase retrieval algorithm will be helpful here. Suppose we have obtained one of the POM ${K}_{2}(x,y)$ and the correct fractional order. By this we mean that we can obtain the amplitude distribution in fractional Fourier domain, nonetheless fall short of providing another constraint to find ${K}_{1}(u,\nu )$. It should be noted that the exact phase distribution in the input plane where the secret image is placed is essential to the correct recovery of the target image [19]. If the amplitude distribution in fractional Fourier domain and an arbitrarily selected phase key are set as two constraints in two planes in the second iterative amplitude-phase retrieval process, the MSE between the amplitude distribution in fractional Fourier domain and its approximates after 50 iterations is demonstrated in Fig. 12(a).The final decrypted amplitude image is illustrated in Fig. 12(b). As analyzed in Section 2, it is impossible for an attacker to obtain the two POMs, i.e., ${K}_{1}(u,\nu )$ and ${K}_{2}(x,y)$, by using a fake image (an image different from the plaintext) and two arbitrarily chosen RPMs based on the proposed algorithm. For the sake of brevity, known-plaintext attacks based on fake images are not demonstrated here.

As we remarked in section 1, one of the best thing about the second method is that we can use several parameters, i.e., the fractional order, the bifurcation parameter and the initial value as the decryption keys. In the following, we encrypt and decrypt the secret image, Lena, based on the second algorithm. The CRPM used for encryption is produced with the initial value ${x}_{0}=0.12$ and the bifurcation parameter $\mu =3.94$. The phase distribution of the CRPM is demonstrated in Fig. 13(a).Figure 13(b) depicts the encrypted result with the same fractional order, $\alpha =0.6$. The curves of MSE and CC versus number of iterations after 500 iterations are presented in Fig. 14.Compared with the curves presented in Fig. 7, it is obvious that the number of iterations will increase for the new algorithm to achieve a low MSE/CC where a CRPM is used as a fixed phase key. Likewise, the MSE decreases very fast when the number of iterations goes from 1 to 100. The MSE values corresponding to the numbers of iterations, 1, 5, 25, 100 are about $\text{7}\text{.7}\times {\text{10}}^{\text{-2}}$, $1.5\times {10}^{-3}$, $\text{3}\text{.2}\times {\text{10}}^{\text{-3}}$ and $\text{7}\text{.6}\times {\text{10}}^{\text{-4}}$, respectively. The CC values corresponding to the iteration numbers above are 0.4752, 0.8542, 0.9668 and 0.9910, respectively. It is obvious that this new algorithm has a slower convergent rate when compared with the previous algorithm without using CRPMs.

Four decrypted images with respect to the numbers of iterations, 10, 50, 150 and 500, are presented in Figs. 15(a)–15(d) with the corresponding MSE values of $\text{7}\text{.6}\times {\text{10}}^{\text{-3}}$, $\text{1}\text{.4}\times {\text{10}}^{\text{-3}}$, $\text{5}\text{.3}\times {\text{10}}^{\text{-4}}$ and $\text{2}\text{.0}\times {\text{10}}^{\text{-4}}$, respectively. In fact, the decrypted image obtained after 150 iterations is sufficiently clear.

Now we turn to the problem of information disclosure. The amplitude image resulting from performing FrFT to the encrypted result $E(u,\nu )$ with the correct fractional order of $-0.6$, is shown in Fig. 16(a).Figure 16(b) shows the image reconstructed from the encryption key, $C(x,y)$ with the correct fractional order. The MSE between the decrypted image and the target image versus the fractional order is shown in Fig. 17.

For the two methods proposed in this paper, in fact, the encryption processes are different but their decryption processes are almost the same. Thus, their security features are extremely similar. The only change here from what we found in the first method is that the parameters of CRPM provides extra level of security. Figures 18(a) and 18(b) shows the wrong decrypted images with the correct encrypted image $E(u,\nu )$ but incorrect CRPMs, where one parameter of the CRPM $C(x,y)$ is changed very slightly while another remains fixed.

## 5. Conclusion

In summary, the method for optical image hiding using phase retrieval algorithm based on iterative nonlinear DRPE is presented in this paper, by which a secret image can be encoded into two POMs. Simulation results indicate the strong convergence of our algorithm. No post-processing of the two POMs or digital inverse computation is involved in our proposed method, as well as the problem of undesired information disclosure. The optical DRPE scheme in fractional Fourier domain can be directly applied for decryption without the need of phase recording. We also proposed a modified encryption method to reduce the load of key transmission without significantly decreasing the quality of image recovery based on modified phase retrieval algorithm and logistic map. The POM obtained in the encryption process is stored as encrypted result while the fractional orders, the bifurcation parameter and the initial value for the logistic map that provide extra security are kept and transmitted as keys.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61205006), the Asian Office of Aerospace Research & Development under Grant No. AOARD 134106 and the State Scholarship Fund of the China Scholarship Council (CSC) under Grant No. 201308330343.

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