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A novel approach for silhouette-free image encryption based on interference is proposed using discrete multiple-parameter fractional Fourier transform (DMPFrFT), which generalizes from fractional Fourier transform. An original image is firstly applied by chaotic pixel scrambling (CPS) and then encoded into the real part of a complex signal. Using interference principle, the complex signal generates three phase-only masks in DMPFrFT domain. The silhouette of the original image cannot be extracted using one or two of the three phase-only masks. The parameters of both CPS and DMPFrFT can also serve as encryption keys to extend the key space, which further enhance the level of cryptosystem security. Numerical simulations are demonstrated to show the feasibility and validity of this approach.
© 2017 Optical Society of America
1. Introduction
As the greater requirement of image sharing in the contemporary society, image security has caused intense interest in different walks of life. Since Javidi et al. [1] employed double random phases encoding (DRPE) to encrypt an image into white stationary noise, optical encryption methods have been playing increasing roles in application of image security due to their parallelism, high speed and wide degrees of freedom [2]. To improve the security of the cryptosystems, DRPE-based configurations have been extended from Fourier transform (FT) domain to various transform domain, including fractional Fourier transform (FrFT) [3], Fresnel transform (FRT) [4], gyrator transform [5], fractional angular transform [6], fractional random transform [7] and discrete multiple-parameter fractional Fourier transform (DMPFrFT) [8,9]. However, these DRPE-based schemes usually require holographic system to record the complex valued encoded ciphertext, which lead the information storage and transmission to be difficult. Besides, the DRPE-based schemes are also demonstrated with low endurance to several attacks because of inherent linearity [10–13]. The first problem can be solved using phase retrieval methods to iteratively encode the original image [14,15], but they come at the expense of time-consuming. To decrease the linearity, permutation-diffusion processes have been employed using different chaotic maps [16–18], but they complicate the optical cryptosystems. Phase-truncated Fourier transforms (PTFTs) can eliminate the linearity of DRPE completely, but the initial scheme is also vulnerable to a specific attack [19]. Much work has also been done to improve the security of PTFTs, but they also make the encoding scheme more complicated [20]. Recently, Zhang and Wang proposed an encryption approach using interference to encode a plain image into two phase only masks (POMs), which benefits from simplicity and non-iterative operation [21]. Much work has been done on interference-based encryption methods [22–25]. However, the silhouette appears even if only one of the two POMs is utilized during the verification process. Different approaches have been developed to overcome the silhouette problem [26–29], one of which encodes the original image into three POMs with one generated randomly and the other two generated analytically [30]. However, despite significant progress, higher security remains a grand challenge in the image optical encryption using interference.
In this paper, we present a new interference-based image encryption method using CPS and DMPFrFT to generate three POMs. Any information relative to the original image cannot be observed even one or two of the POMs are used for decryption. Furthermore, the key space are improved significantly due to the introduction of CPS and DMPFrFT compared with the traditional optical encryption using interference [21–23].
2. Principle
In an encryption system using interference, an original image o(u, v) to be encrypted is always transferred to a complex value image o′(u, v) [21] firstly as follows.
where o(u, v) is a nonnegative distribution, and M(u, v) is a random phase distribution in the range of [0, 2π].To further improve the security of the system, an operation P{a0, λ, t} [8] of chaotic pixel scrambling (CPS) is firstly applied to the original image, and Eq. (1) can then be rewritten as
where a0 is the initial value located in the range of [0, 1], λ is the coefficient of the map located in the range of [3.57,4], and t is truncated position.To overcome the inherent characteristics of the silhouette problem in the interference-based encryption system, the encrypted image can be encrypted into three POMs [30] of M1(x, y), M2(x, y) and M3(x, y), where M1(x, y) and M2(x, y) are generated analytically, and M3(x, y) is generated randomly in the range of [0, 2π]. But unlike the previous methods are done in the domain of FRT [21], FT [22] or FrFT [23], our interference is done in the DMPFrFT domain. In terms of the FRT, FT or FrFT, the DMPFrFT has more parameters to provide larger key space and more diversified signal expression, which has been shown better performance and security in image encryption in conventional encryption method [8,9]. When the three POMs are illuminated by a plane wave simultaneously, interference in the image plane can be expressed as
where F(•) stands for DMPFrFT operation with parameters of (ML,MR; αL, αR; mL, nL; mR, nR), (αL, αR) stands for fractional order, (ML, MR) stands for periodicity, (n′L, n′R) stands for vector parameter, and n′ is defined asand m = (m0, m1, m(M-1))∊ℤM, n = (n0, n1, n(M-1))∊ℤM, and M is arbitrary integer of > 2.Let us rewrite Eq. (3) so that to obtain a relationship defining M2(x, y) by M1(x, y), M3(x, y) and o′(m, n) as follows
Since M2(x, y) is phase-only elements, we can obtainwhere * represents the conjugate operation.Combining Eqs. (3) and (6), we can achieve the phase distributions of M1(x, y) and M2(x, y) as
where arg[•] represents the phase angles of the complex function and abs[•] represents the complex modulus.In our proposed optical encryption system, the encryption is processed digitally in a computer, while the decryption can be performed digitally or optically. Figure 1 shows the optoelectronic setup for the proposed decryption procedure in DMPFrFT domain. In the setup, SLM1 is employed to generate the summation of the three POMs. SLM2 and Lens are employed to carry out the DMPFrFT [9]. A laser is collimated and expanded by a beam collimator & expander (BCE) to generate coherent parallel light beams. The light is modulated by SLM1, and then transformed by SLM2 and Lens. The scrambled image can be recorded by CCD and transferred to a computer. The decrypted image can be finally obtained after the inverse CPS is digitally implemented in the computer.
Fig. 1 Optoelectronic schematic for decryption. BCE, beam collimator and expander; SLM1 and SLM2, spatial light modulator.
In our algorithm, the three POMs can be sent to three different holders to achieve high security. Even if the attackers illegally obtain one or two of the POMs, they cannot retrieve any information about the original image, including its silhouette. Especially, the parameters of both CPS such as initial value a0, coefficient λ, truncated position t, and DMPFrFT such as the periodicities (ML, MR), transform orders (αL, αR) and vector parameters (mL, nL; mR, nR) can also be used as the encryption keys. When the three POMs are illuminated by the plane wave with correct parameters, the plaintext can be achieved in the output plane. To assess the quality of the decrypted image, we use the normalized mean square error (NMSE) [9] as follows
where Io(x, y) is the values of the original image and ID(x, y) is that of the decryption image, and p and q are the size of the images.3. Results
Numerical simulations have been done to validate the proposed method. Figure 2(a) shows an original image with 256 × 256 pixels size and 256 Gy levels, and Figs. 2(b) and 2(c) show its corresponding POMs of M1 and M2 generated using the proposed encryption approach. The results indicate that the image can be completely encoded, and no any valuable information related to the image can be recognized. The POM of M3 is also generated using CPS, and its distribution is shown in Fig. 2(d). During the encryption process, the parameters of CPS are set as {a0, λ, t} = {0.241, 3.95, 3500}, and those of DMPFrFT are set as (αL,αR; ML,MR) = (0.35, 0.74; 13, 22). The vector parameters (mL, nL) and (mR, nR) are 1 × 13 and 1 × 22 random vectors that contain independent integer values, respectively. Figure 2(e) shows the scrambled-image recorded in CCD and Fig. 2(f) shows the corresponding decrypted image when the correct keys are applied. The decrypted image can be identified obviously without any doubt, and the corresponding NMSE value is 7.9384 × 10−27. But it should be noted that since DMPFrFT is a periodic function, the image can be retrieved with another periodicity when it is fractional transformed with a particular periodicity.
Fig. 2 (a) The original image, (b)-(d) the POMs of M1, M2 and M3, (e) the scrambled-image recorded in CCD, (f) the decrypted image.
Next, we illustrate the performance of the security achieved by our proposed method. Figure 3 show the influence of the deviation of fractional order in DMPFrFT on the decrypted image. The results indicate show that our approach is highly sensitive to the fractional order. Any slightly deviation, almost (1 × 10−7, 0.5 × 10−7) from (αL, αR), can make the decrypted image unrecognized. But for the approach using FrFT [23], the deviation from the fractional orders should be ≥0.08.
Fig. 3 The decrypted image with (a) (αL + 1 × 10−7, αR) and (b) (αL, αR + 0.5 × 10−7), (c) the curve of NMSE with incorrect fractional order.
Figures 4 and 5 show the decrypted image extracted by wrong periodicity of (ML, MR) and vector parameters (mL, nL; mR, nR), respectively. The same analyses are also performed on the parameters of CPS, and the extremely sensitive results are shown in Figs. 6-8. It’s no doubt that the information about the original image can hardly be extracted when any error occurred in the parameters of DMPFrFT and/or CPS. All those enable our approach to realize higher security hierarchy in comparison to the conventional approaches.
We further demonstrate the silhouette-free capability of our approach using only one or two of the three POMs during the decryption process, and summarized their results in Fig. 9. The corresponding NMSE values are 0.7843, 0.7809, 0.7816, 0.7839, 0.7792 0.7832 respectively. All the retrieved images show fully noise, and no silhouette information leave. Noted that all the parameters of the DMPFrFT and CPS are selected randomly. We have tried ten different combinations of the parameters for all the above analyses and can obtain the same results.
Fig. 9 The decrypted image with (a) M1, (b) M2, (c) M3, (d) M1 and M2, (e) M1 and M3, and (f) M2 and M3.
4. Conclusion
In conclusion, we have developed an interference-based image encryption method using CPS and DMPFrFT. The silhouette problem has been thoroughly removed using the summarization of two POMs generated analytically and one POM generated randomly. In our scheme, the original image is firstly scrambled and encoded into a complex signal, and then encoded into the three POMs using DMPFrFT. Numerical results show that our approach has extremely sensitivity to the keys. Of course, we realize this at the cost of storage space for the use of three POMs. However, in contrast to the approaches used two POMs together with phase retrieval algorithm [31], our approach can reduce computational load and time efficiently. Thus, we hope that our new method will be highest security and most practical in applications of interference-based image encryption methods.
Funding
National Natural Science Foundation of China (61377009); Major National Scientific Instrument and Equipment Development Project of China (2013YQ290489); China Scholarship Council (201506685053); Heilongjiang Science Foundation of China (F201411); and Fundamental Research Funds for the Central Universities of China.
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