We present a technique of nonlinear image encryption by use of virtual optics. The image to be encrypted is superposed on a random intensity image. And this superposed image propagates through a nonlinear medium and a 4-f system with single phase key. The image is encrypted to a stationary white noise. The decryption process is sensitive to the parameters of the encryption system and the phase key in 4-f system. This sensitivity makes attackers hard to access the phase key. In nonlinear medium, optically-induced potentials, which depend on intensity of optical wave, make the superposition principle frustrated. This nonlinearity based on optically induced potentials highly improves the secrecy level of image encryption. Resistance against attacks based on the phase retrieval technique proves that it has the high secrecy level. This nonlinear image encryption based on optically induced potentials is proposed and demonstrated for the first time.
© 2011 OSA
Optically-induced potentials are generated by propagating beams that create refractive index changes in the host medium. Phenomena such as plasma filamentation and self-focusing of femtosecond pulses are the typical examples. Digital reconstruction of optically-induced potentials is proposed using nonlinear digital holographic technique . In ref , complex field exiting out of a nonlinear medium is recorded using the technique of phase-shifting digital holography [3, 4] and the complex field numerically back-propagates by nonlinear Schrödinger equation. It’s a new useful method in image and signal processing.
Image processing based on optical technology has shown great potential in the field of information security. Many optical encryption schemes were proposed in the past decade [5–9]. The most extensively studied optical security scheme  proposed by Refregier and Javidi uses the double random phase masks to encode an image into a stationary white noise. This method has been extended to fractional Fourier domain  and Fresnel domain .
In recent years, virtual optics is widely used in image encryption in which both encryption and decryption were done digitally [10–12]. Namely, virtual optics digital encryption and decryption have become the main practical technique in optical information security. This method can easily control the phase mask and the optical wave property. Compared with traditional optical setup, there is no need to fabricate physical phase mask which means that the phase mask key can be easier to change; the alignment problem of two or more phase masks can be easily tackled.
The conventional double random phase encoding (DRPE) method is susceptible to chosen plaintext attack (CPA) and known plaintext attack (KPA). The CPA firstly done by Carnicer et al.  indicated the vulnerability of the conventional DRPE scheme. CPA requires the attacker to control the system to encrypt particular image. Peng et al. proposed KPA on conventional DRPE scheme. With the help of the phase retrieval technique, both the random phase keys can be recovered . Frauel et al. proposed an analytical method to access the exact keys with only two known plaintext images . Liu et al. propose a hybrid heuristic algorithm to improve KPA on conventional DRPE scheme, making KPA more practical and effective .
For conventional DRPE scheme, cryptanalysis [14, 15] indicates that linearity is the main weakness of this system. In this paper, we propose and demonstrate nonlinear image encryption based on optically induced potentials for the first time. It makes superposition principle frustrated and results in hiding of plaintext by superimposing a random intensity image on the plaintext. KPA is in vain to this method.
2. Principle of optically-induced-potential image encryption
The schematic of image encryption system is illustrated in Fig. 1 . The image to be encrypted is normalized firstly. Random distributed image which acts as intensity key is superimposed on the plaintext. The intensity key can be expressed as r(x, y), where r(x, y) is an independent random function. The treated plaintext image is uploaded on the spatial light modulator (SLM) in plane P0. The SLM is illuminated by a polarized plane wave and the input field propagates through the nonlinear medium. The field exiting out of the nonlinear medium propagates through a 4-f system with single phase key. Two lenses L1 and L2 with same focal length form the 4-f system. Random phase masks (RPM) M is placed in plane P2. Complex transmittances of the phase mask can be expressed as exp(i2πp(x, y)), where p(x, y) is an independent random function that uniformly distributed in [0, 1]. The complex field in plane P3 is recorded by phase-shifting holography.
For the propagation in nonlinear medium, we consider the case of scalar wave under paraxial approximation. In this condition, the propagation can be described by the nonlinear Schrödinger equation [1, 2]:17,18]:1, 2] in which diffraction and nonlinear operators act individually in a small propagation distance dz. In fact, the diffraction process and the nonlinear process happen simultaneously. But when the propagation distance dz is very small, this approximation will have very small error.
The complex-amplitude of output field of the nonlinear medium can be obtained by:
The complex field in plane P3 is
In procedure of decryption, with correct phase key and correct parameters of the system, the field in plane P3 back-propagates through the system.
The backward propagation in the nonlinear medium is also an initial-value problem. The output field is the starting point. The amplitude of field in plane P0 can be obtained by:
And the decrypted result could be obtained with an intensity key.
The encryption and decryption procedure could be summed up as follows. The superposition of plaintext and intensity key form input field of the encryption system. The encryption procedure is the forward propagation of the input field. And the decryption procedure is the backward propagation of ciphertext with correct phase key and correct parameters of the system to get the input field in plane P0. With a correct intensity key, correct plaintext could be obtained from the input field. During propagation in nonlinear medium, the wavefront of the intensity key and the plaintext would interact with each other. The output field of nonlinear medium in plane P1 is not the sum of individual output field of the intensity key and the plaintext because of this interaction. This property makes attackers hard to retrieve intensity key.
3. Results and discussions
We take out numerical experiments to verify feasibility and effectiveness of this method. The pixel size on SLM in plane P0 is 10μm*10μm. In this situation, paraxial approximation could be satisfied. The Lena image with 256×256 pixels shown in Fig. 2(a) is used as plaintext. Zeros is padded to extend the size of the image to 512×512 and a random intensity image is superimposed on the plaintext. The length of the photorefractive crystal Fe:LiNbO3 is 20cm and parameters of the crystal are Epv=−27×105V/m, ne=2.20, γeff=30.9pV/m, at the wavelength of λ=514.5nm. The intensity of the treated Lena image is normalized with the background intensity. The optical wave in the crystal is e-polarized. The step length dz is 200μm. After the propagation of the nonlinear crystal, the real-part/image-part of complex-amplitude of the field exiting out of the nonlinear medium in plane P1 is shown in Fig. 2(b)/ (c). The real-part/image part of the ciphertext is shown in Fig. 2(d)/ (e). Decrypted image with correct keys and correct parameters is shown in Fig. 2(f). The correlation coefficient (CC) between the plaintext and the decrypted image is 1.0000 meaning that the image can be decrypted without any distortion. The correlation coefficient is defined as CC(x,y)=cov(x,y)/(σxσy), where cov(x,y) is the cross-covariance of two variables and σxσy is the product of two variables’ standard deviations.
The parameters of the system should also be exactly correct in order to get correct decrypted image when the phase key in plane P2 and the intensity key in plane P0 are exactly known. In decryption procedure, the parameter of nonlinear crystal should be exactly correct, or the optically-induced potentials would go wrong which makes the decrypted result incorrect. When γeff increases by 0.0001%, the decrypted image is shown in Fig. 3(a) . The decrypted image is added with noise. When γeff increased by 0.0005%, the decrypted image is shown in Fig. 3(b), the useful information is totally blurring. When γeff increased by 0.001%, the decrypted image is shown in Fig. 3(c), the decrypted image becomes a random noise. The decrypted result is sensitive to nonlinear parameter γeff.
In a small propagation distance in the nonlinear medium, there are linear diffraction process and nonlinear process. The decrypted result is also sensitive to the parameter of linear process. When the wavelength λ increases by 0.0001% in linear process while the nonlinear parameters are kept exactly correct, the decrypted image is shown in Fig. 3(d). The decrypted image is added with noise. When λ increased by 0.0005% in linear process, the decrypted image is shown in Fig. 3(e), the useful information is totally blurring. When λ increased by 0.001%, the decrypted image is shown in Fig. 3(f), the decrypted image becomes a random noise. The linear diffraction parameter is also critical to the decrypted result.
To investigate the decryption sensitivity to parameters of the system when the phase key in plane P2 and the intensity key in plane P0 are exactly known, we calculate the dependence of CC on the nonlinear coefficient and wavelength λ in linear diffraction process. We define nonlinear coefficient as:
The dependence of CC on δγ/γ and the dependence of CC on δλ/λ are shown in Fig. 4(a) and Fig. 4(b), respectively. When CC<0.2, the decrypted image cannot be distinguishable. When the deviation of γ is larger than 0.001%, the decrypted image would be undistinguishable. The insets show the zoomed figures. It’s clear that the dependence of CC on nonlinear coefficient γ or linear wavelength λ is highly sensitive.
We investigate decrypted results when a random noise is superimposed on the phase key in plane P2. The random noise can be expressed as Amexp(i2πq(x, y)), where q(x, y) is an independent random function that uniformly distributed in [0, 1], Am is the amplitude of the noise. A random noise whose amplitude Am is 0.00001% to the amplitude of the phase key (the amplitude of the phase key is 1) is superimposed on the phase key in plane P2. The decrypted image with this noise-added phase key is shown in Fig. 5(a) .the decrypted result is added with noise. When the amplitude of the noise increases to 0.0000005,the quality of the decrypted image decreased dramatically(see Fig. 5(b)). When the amplitude of the noise increases to 0.000001, the decrypted image cannot be distinguishable (see Fig. 5(c)). When the amplitude of the noise increases to 0.000005, the decrypted image becomes a noise (see Fig. 5(d)). We also investigate decrypted results in the absence of the nonlinear medium when a random noise is superimposed on the phase key in plane P2. While the parameter of γeff equals zeros (in the absence of nonlinear medium), the encryption system becomes a linear system. Figure 5(e), (f), (g), and (h) respectively show the decrypted images in the absence of the nonlinear medium while random noises with different amplitude of 0.01, 0.05, 0.1, and 0.5 are superimposed on the phase mask M. When the amplitude of the noise increases to 0.1, the decrypted image is still distinguishable (see Fig. 5(g)). The comparison of the decrypted results with noise-added phase key shows that phase encoding in the presence of the nonlinear medium is more sensitive to phase changes than in the absence of the nonlinear medium.
In order to analyze the nonlinearity of this system, we investigate the situation of superposition of two ciphertexts. For linear system, the output of superposition of two (or more) weighted input functions is the superposition of the weighted individual outputs. In our proposed method, the condition is not satisfied because nonlinear phase change depends on the intensity of optical wave. When the superposition of two plaintexts (ciphertext) becomes a new plaintext (ciphertext), nonlinear phase changes are not the same as they do in the situation of single plaintext (ciphertext). Figure 6(a) and Fig. 6(b) show a pair of ciphertext and corresponding decrypted plaintext. Figure 6(c) and Fig. 6(d) show another pair of ciphertext and corresponding decrypted plaintext. The decrypted result with the sum of the two ciphertexts with weights of 0.5 is shown in Fig. 6(e). In Fig. 6(f), the linear superposition of two plaintexts is shown. Compared with Fig. 6(f), useful information of the plaintext in Fig. 6(e) vanishes. The frustration of superposition principle means the encryption system is nonlinear. Based on this property, hiding of plaintext could be realized by superimposing a random intensity image on the plaintext.
For known plaintext attack, proposed attacker knows ciphertext (shown in Fig. 7(a) ) and corresponding binary plaintext (shown in Fig. 7(b)). The binary plaintext would be easier to achieve good attack result . Phase retrieval technique is used to calculate the phase key, ignoring the nonlinear crystal and the intensity key. The plaintext retrieved by hybrid input–output algorithm  is shown in Fig. 7(c). The attacker uses the retrieved phase key to decrypt another ciphertext (shown in Fig. 7(d)). The decrypted result is shown in Fig. 7(e), while correct plaintext of this ciphertext is shown in Fig. 7(f). The decrypted image with the retrieved phase key doesn’t contain any information of the correct plaintext. To access phase key using phase retrieval technique, one should get the correct complex-amplitude in plane P1. In our system, the complex-amplitude of the field in plane P1 is the output field of the nonlinear crystal. The attacker cannot obtain the distribution of complex-amplitude in plane P1 because of the existence of the intensity key and the nonlinear medium. Phase retrieval technique which works in the linear transform systems doesn’t work in nonlinear transform systems. In our nonlinear encryption system, the wavefront would be distorted by the intensity of field. The wavefront of the intensity key and the plaintext interact with each other during propagation in nonlinear medium. Phase retrieval technique to recover the phase key will not work. All the attacks based on phase retrieval technique will fail.
In summary, we proposed an optical encryption method with nonlinear medium. The plaintext overlapping with a random intensity key is the input image of the encryption system. The input image propagates through a nonlinear medium and the field exiting out of the nonlinear medium passes a 4-f system with single phase key. Decryption procedure is the backward propagation of ciphertext. The decryption is sensitive to the parameters of the system. The frustration of superposition principle of the encryption system is demonstrated. The system can resist attacks based on phase retrieval technique such as KPA because of its nonlinearity and the random intensity image superposed on the plaintext. The simulation results have shown this encryption method based on optically-induce potentials has high secrecy level.
This work is supported by the National Natural Science Foundation of China (11074311, 10804131, and 10874250), the Fundamental Research Funds for the Central Universities (2009300003161450), and the Guangdong Natural Science Foundation.
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