1. Introduction

Optical technology has been widely used in information encryption and decryption application, owing to its multiple parameters and parallel processing ability. In the past decade, various algorithms and systems about optical image encryption and security have been proposed [1–4]. Refregier and Javidi [1] proposed a method that uses double random phases to encrypt an image to stationary white noise. Liu et al. [5] and Zhang et al. [6] extended this method into the fractional Fourier domain. The security of double random phase encoding has also been thoroughly analyzed and a few weaknesses have started to appear [7–9]. In order to make it easy to decode the encrypted image with intensity-sensitive detectors,Wang et al. have also proposed a method to encrypt the original image into a phase-only mask in the Fourier plane of a 4 f optically system [10]. Li et al. improved this method to encrypt the image into the mask in the input plane [11]. Recently, it has also been expanded to encrypt multiple images [12]. However, all of the above-mentioned works used the iterative algorithm to encode the original image into the pure phase diffraction elements (DPEs). Furthermore, the basic ideas of these methods are based on the diffraction principle. The encryption methods based on the interference principle have also been investigated. Javidi and Nomura used a digital hologram to store an encrypted image and the decryption key [13], and Nishchal et al. proposed a method of securing information using fractional Fourier transform in digital holography [14]. Meng et al. proposed a hybrid cryptosystem, in which one image is encrypted to two interferograms with the aid of double random-phase encoding (DRPE) and two-step phase-shifting interferometry (2-PSI) [15]. Only recently, Zhang et al. [16] proposed a new encryption method that an image is encoded into two pure phase masks based on the optical interference principle. The encryption method is quite simple and does not need the iterative algorithm and the used encryption keys have high sensitivity. They encode information of an image into two pure phase masks by using only one wavelength and the two pure phase masks must have identical distances from the output plane, which make invaders easy to obtain the distance key and the wavelength key by testing repeatedly.

In this article, we proposed an encryption and verification method, which is also based on optical interference. In the new method, information of two different images are encoded into three pure phase diffraction elements (DPEs) by using two different illuminating wavelengths, respectively. The three DPEs have different distances from the output plane. The two illuminating wavelengths and the three distances can be used as encryption keys, which significantly increase security degree. For simple, we only encode image information into three DPEs in this article, more than three DPEs can also be employed to enhance performance of the encryption system. The encryption process is implemented in computer digitally and the verification process can be realized through optical system or digitally.

2. The verification optical system

The verification optical system is schematically shown in Fig. 1 . There are two methods for verification process. In Fig. 1(a), three coherent parallel light beams are modulated by three diffractive phase elements (DPEs) and then are combined by two half-mirrors (HM1 and HM2). Thus three beams interfere with each other on the output plane and generate the image encrypted in the three DPEs. Notice that three DPEs comprise two kinds of encrypted information for two different wavelengths, that is, using two groups of incident light with different wavelengths will generate two different images on the output plane. The two wavelengths can be used as encryption keys. In verification process (or, decryption process), an image detector such as Charge Coupled Device (CCD) can be used to accept image on the output plane. In order that three light beams through three DPEs have appropriate field intensities on the output plane, we use an optical attenuator (OA) on the optical path of DPE3. The distance between DPE1 and output plane $l_1$ is set to be the same as the distance between DPE2 and output plane $l_2$, but the distance between DPE3 and output plane $l_3$ is different from $l_1$ and $l_2$. So these different distance parameters can also be used as encryption keys and enhance secrecy degree. A simpler verification system is shown in Fig. 1(b). Three coherent light beams transmit in different directions through three DPEs, but the three light beams will intersect with each other on the output plane and then generate the encrypted image. In this case, if angles between three light beams α and β are small, the complex field distributions of light beams through DPE1 and DPE3 on the output plane will only need to multiply $\exp (i\frac{2\pi }{\lambda }\alpha x)$ and $\exp (-i\frac{2\pi }{\lambda }\beta x)$, here, x represents vertical coordinate on the output plane and λ is wavelength of incident light beam.

 

Fig. 1 Schematic of the verification optical system

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The encryption process should be implemented digitally, while the decryption process can be implemented optically or digitally. For highly secure verification, three DPEs can be assigned to three different important users, and the system parameters (three distance parameters and two wavelength parameters) can also set as encryption keys, which not only can be fixed in the verification system, but also set as passwords for different users. As a result, the encrypted image can be obtained at the output plane only when three correct users put three DPEs in the verification system and input correct system parameters.

3. Digital encryption algorithm

The encryption problem is equal to separate information of an image into three phase diffractive elements (DPEs). In the present method, we use two wavelengths to encrypt the image information (here, represented by ${\lambda }_1$ and${\lambda }_2$). Firstly, we use the first wavelength ${\lambda }_1$.

If the encrypted image is an intensity distribution $o_{\lambda 1}(m,n)$, a new complex distribution can be constructed by adding it with a random phase distribution,

If the relationship between distance from DPE1 to the output plane and distance from DPE2 to the output plane is $l_1=l_2=l$ and set the phase distribution $\mathrm{\Phi }{3}_{\lambda 1}$ of DPE3 as a random phase distribution between 0 and $2\pi$, an equation can be obtained after a simple deduction,

Since module of the left portion of the Eq. (6) is equal to 1, we have

At last, we can achieve the phase distributions of DPE1 and DPE2 as

According to the same principle, the phase distributions of three DPEs for incident light with wavelength ${\lambda }_2$ can be obtained. Notice that the three phase distributions $\mathrm{\Phi }{1}_{\lambda 2}$,$\mathrm{\Phi }{2}_{\lambda 2}$and $\mathrm{\Phi }{3}_{\lambda 2}$ correspond to an interference intensity distribution $o_{\lambda 2}(m,n)$ on the output plane that is different with the intensity distribution $o_{\lambda 1}(m,n)$ for the first wavelength${\lambda }_1$.

Now, we have three phase distributions for ${\lambda }_1$ and three phase distributions for${\lambda }_2$, but only three DPEs can be used to generate these phase distributions. We express the height distribution of surface-relief structure of three DPEs as$h_1$, $h_2$ and$h_3$, respectively. For DPE1, the phase distribution corresponding to incident wavelength ${\lambda }_1$ can be expressed as

4. Numerical simulations

Computer simulations are carried out to show validity of our proposed new method. The images to be encrypted are two $256\times 256$ pixel gray images which correspond to incident wavelengths ${\lambda }_1$ and ${\lambda }_2$, respectively. As shown in Figs. 2(a) and 2(b), two images comprise different letters. The sizes of two images are both $5cm\times 5cm$, and two wavelengths of the illuminating light are set as ${\lambda }_1=633nm$ and ${\lambda }_2=514nm$, respectively. The three distances are $l_1=l_2=20cm$ and $l_3=40cm$. According to the encryption method mentioned above, we can obtain two groups of phase distributions for two incident wavelength, then the height distributions of three DPEs can be achieved based on Eq. (11). Here, the largest height of relief structures of three DPEs is set as $6\mu m$. The computed height distributions of three DPEs are displayed in Figs. 3 . It is obvious that no information about the original images can be found in height distributions of three DPEs.

 

Fig. 2 Encryption results. (a) The first original image for encryption, (b) the second original image for encryption, (c) and (d) two reconstructed images with correct encryption keys, (e) and (f) two reconstructed images with incorrect encryption keys.

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Fig. 3 Height distributions of three encrypted DPEs

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The reconstructed images with correct height distributions of three DPEs and right system parameters are displayed in Figs. 2(c) and 2(d), respectively. It can be seen that the two primary images have been retrieved well, but only some fluctuations exist. These fluctuations can be eliminated by increasing values of P and Q in Eq. (11), i.e., enlarging the largest heights of three DPEs. However, too large height will make fabrication of DPEs difficult. If we set the height distribution of DPE3 as incorrect values, the reconstructed images for two correct wavelengths are shown as Figs. 2(e) and 2(f). It is clear that we cannot observe any useful information from the reconstructed images.

To evaluate the reliability of the encryption method quantitatively, the relative error (RE) between the two reconstructed images and two original images is introduced as

In the proposed method, the two wavelengths of the illuminating light ${\lambda }_1$ and ${\lambda }_2$ can be used as encryption keys. We have calculated the dependence relationship of RE on the two wavelengths, which is shown in left portion of Fig. 4(a) . It can be seen in left portion of Fig. 4(a) that only when both of the two illuminating wavelengths have correct values, RE has the smallest value $\text{RE}=0.0095$; value of RE will sharply rise with increasing of wavelength error.

 

Fig. 4 (a) Dependence of RE on wavelength difference $\mathrm{\Delta }{\lambda }_1$ and $\mathrm{\Delta }{\lambda }_2$, (b) decryption result with $\mathrm{\Delta }{\lambda }_1=1\times {10}^{-4}nm$, (c) decryption result with $\mathrm{\Delta }{\lambda }_2=1\times {10}^{-4}nm$.

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More precise simulation is done to evaluate the evolution of the smallest RE value with $\mathrm{\Delta }{\lambda }_1$, as shown in right portion of Fig. 4 (a). It can be seen from right figure of Fig. 4(a) that RE quickly increases with $\mathrm{\Delta }{\lambda }_1$even in magnitude order of ${10}^{-4}nm$. If the first illuminating wavelength has a tiny difference from the correct value, e.g., $\mathrm{\Delta }{\lambda }_1=1\times {10}^{-4}nm$, which corresponds to$RE=0.25$, the first reconstructed image will not display any useful information about original image, as shown in Fig. 4(b). The same phenomenon can appear for the second illuminating wavelength ${\lambda }_2$, whose corresponding reconstructed image is shown in Fig. 4(c) for $\mathrm{\Delta }{\lambda }_2=1\times {10}^{-4}nm$. It can be proved that the decrypted image cannot be distinguished with the naked eye for $RE>0.2$. The wavelength sensitivity of the proposed encryption method is about $1\times {10}^{-4}nm$.

The distances between each DPE and the output plane can also be used as encryption keys. In our proposed method, the three distances have the relationships $l_1=l_2=l$ and $l_3\ne l$. In order to validate effect of the two distance encryption keys, we have also calculated the influence of change of l and $l_3$ on RE, as shown in left portion of Fig. 5(a) . When $l=20cm$ and $l_3=40cm$, RE has smallest value 0.0095. If l and $l_3$ differs from correct value little, value of RE will significantly increase. More precise simulation is done to evaluate the evolution of the smallest RE value with $\mathrm{\Delta }l$, as shown in right portion of Fig. 5 (a). It is obvious that value of RE quickly rise increases with $\mathrm{\Delta }l$even in so small varying scope. The same evolution of the smallest RE value is obtained for $\mathrm{\Delta }{l}_3$. Figures 5(b) and 5(c) show the reconstructed images for the case of $\mathrm{\Delta }l=2nm$ and $\mathrm{\Delta }{l}_3=2nm$, respectively. It is obvious that the two reconstructed images are entirely different from two original images, that is, tiny difference of distance will fail to decrypt correct information. Further simulation can obtain a conclusion that, when$RE>0.2$, there is a failure to distinguish the decrypted image. The distance sensitivity of the proposed encryption method is about $2nm$.

 

Fig. 5 (a) Dependence of RE on distance difference $\mathrm{\Delta }l$ and $\mathrm{\Delta }{l}_3$, (b) decryption result with$\mathrm{\Delta }l=2nm$, (c) decryption result with$\mathrm{\Delta }{l}_3=2nm$.

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The high wavelength and distance sensitivity will cause great difficulty in copying the encryption system, which make the proposed encryption system have high security degree. Meanwhile, a more precise mechanical supporting system for determining distance parameters l and $l_3$ is needed to implement the verification process. It can also be concluded from further detailed simulation that transversal displacement of three DPEs will significantly influence quality of reconstructed images. For example, a transversal displacement with one-tenth pixel of DPE1 (which corresponds to about $20\mu m$) can cause an increase of relative error for $\mathrm{\Delta }RE=0.1$. Therefore, precise control of the transversal positions of three DPEs is also very important for practical optical verification system. Finally, we also calculate influence of surface-relief deviation of three DPEs on reconstructed images. Tolerance scope of surface-relief height of three DPEs is about $\pm 0.4\%$of surface-relief height. The scope lies in the region of state of the art of optical fabrication and means that the optical encryption process can be implemented conveniently.

5. Conclusion

In conclusion, we proposed a encryption and verification method based on interference principle. The encryption process is digitally implemented on computer and the verification process can realize digitally and optically. Two different original images are encrypted into three DPEs by using two illuminating wavelengths, respectively. The three DPEs comprise encrypted information and are assigned to three different users as encryption keys. The three DPEs have different distances from the output plane. The two wavelength parameters and the three distance parameters can also be used as encryption keys. In verification process, three correct users put their own DPE in the verification system and input two wavelength parameter and three distance parameter. The verification will set correct positions of three DPEs and illuminating lights, and the correct encrypted images can be generated on the output plane only when all of encryption keys are correct. Because of multiple encryption keys in our proposed method and high sensitive of these encryption keys, the interloper will difficulty in copying the decryption system or passing through the verification system. The simulation results have shown the validity of the proposed method.