Single-beam image encryption using spatially separated ciphertexts based on interference principle in the Fresnel domain

Optics Communications 333 (2014) 151–158

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Single-beam image encryption using spatially separated ciphertexts based on interference principle in the Fresnel domain Qu Wang a,n, Qing Guo b, Liang Lei a, Jinyun Zhou a a b

School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou, 510006, China Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing, 100094, China

art ic l e i nf o

a b s t r a c t

Article history: Received 10 June 2014 Received in revised form 10 July 2014 Accepted 29 July 2014 Available online 10 August 2014

A new optical security system for image encryption based on optical interference principle and translation property of Fresnel transform (FrT) has been proposed in this article. The algorithm of this proposal is specially designed for single-beam optical decryption and can thoroughly resolve the silhouette problem existing in the previous interference-based scheme. Different from earlier schemes using interference of phase-only masks (POMs), the inverse FrT of primitive image is digitally decomposed into a random POM and a complex field distribution. Information associated with the primitive images can be completely smoothed away by the modulation of this random POM. Through the translation property of FrT, two linear phase-only terms are then used to modulate the obtained random POM and the complex distribution, respectively. Two complex ciphertexts are generated by performing digital inverse FrT again. One cannot recover any visible information of secret image using only one ciphertext. Moreover, to recover the primitive image correctly, the correct ciphertexts must be placed in the certain positions of input plane of decryption system, respectively. As additional keys, position center coordinates of ciphertexts can increase the security strength of this encryption system against brute force attacks greatly. Numerical simulations have been given to verify the performance and feasibility of this proposal. To further enhance the application value of this algorithm, an alternative approach based on Fourier transform has also been discussed briefly. & 2014 Elsevier B.V. All rights reserved.

Keywords: Image encryption Fresnel transform Optical interference

1. Introduction In the past two decades, extensive studies have been carried out to apply coherent optics methods for information security due to their high security level, parallelism and ultrafast processing speed [1–22]. The pioneering work of this field is the double random phase encoding (DRPE) technique, proposed by Refregier and Javidi in 1995 [2]. DRPE enables one to encode a primary image into a stationary white noise by placing two random phase keys in the input and Fourier planes, respectively. The extensions of DRPE in the fractional Fourier domain [3,4] and the Fresnel transform (FrT) [5–9] domain have also been reported. To alleviate the strict optical alignment, the optical DRPE can be also performed by means of a joint transform correlator (JTC) architecture [10–14] or joint fractional Fourier transform correlator (JFTC) [15,16]. However, recent works have demonstrated that DRPE is vulnerable to some special attacks, due to inherent linearity [17–19]. Recently, Zhang and Wang introduced a very simple interferencebased method to encode a primitive image into two phase-only

n

Corresponding author. Tel.: þ 86 020 8708 4387. E-mail address: [email protected] (Q. Wang).

http://dx.doi.org/10.1016/j.optcom.2014.07.085 0030-4018/& 2014 Elsevier B.V. All rights reserved.

masks (POMs) without any time-consuming iterative computation involved [23]. With the help of the two analytically obtained POMs in the encryption process, authorized users can record the decrypted image directly in the output plane by using an intensity recording device. However, the earlier interference-based encoding (IBE) system has an inherent problem, as silhouette information of the primitive image can be discerned when any one of these masks is employed in the decryption process. To eliminate the inherent silhouette problem in the original IBE schemes, some methods have been reported, but most of them require time-consuming computation for postprocessing or a greater number of POMs [24–26]. With the expectation to resolve this problem at the source, Wang and Zhao introduced a triple-POM-based scheme [27]. In their scheme, an unauthorized user cannot detect the silhouette of the original image using only one POM because the phase information of the inverse Fourier spectrum of the original image is separated and modulated by a random phase factor. Although the implementation of this scheme is very compact, it is still possible to discern the remnant information if an unauthorized user possesses two of the three POMs simultaneously. More recently, we proposed another triple-POMbased scheme to thoroughly resolve the silhouette problem at the source, in which any recognizable information of the secret image cannot be discerned unless the potential attacker acquires all of the

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three POMs [28]. By combining this scheme and POM multiplexing, a multiple-image encryption system has also been constructed [29]. Moreover, recently, we found that polarized light encoding can also be utilized to get rid of the silhouette problem [30,31]. However, in these schemes, beam splitting and many diffraction devices must be used, which means strict optical alignment and system stability are required in such encryption systems. In this article, we will introduce a new IBE scheme for image encryption that can avoid the inherent silhouette problem of the earlier IBE schemes. The algorithm of this proposal is specially devised for single-beam implementation structure, thus no beam splitting is involved during decryption process. Moreover, more decoding keys are introduced to further strengthen the security level of system. During encryption process, the original image is firstly converted into a random phase mask (RPM) and a complex mask. Then, both the masks are respectively modulated by two predesigned phase-only factors. Finally, one can get two complex ciphertexts by carrying out an inverse FrT on the modulated masks. The aforementioned processes must be completed by digital means. To extract the secret information, one just need to properly shift the two complex ciphertexts by certain amounts and modulate them by two phase-only factors in the input plane. Then an FrT with certain propagation parameters, including wavelength and distance, is performed to recover the original image. The translation coordinates of ciphertexts, together with the FrT parameters, can be considered as additional keys of this proposal. A compact optoelectronic setup is suitable to complete the decryption, which only needs a complex SLM to display ciphertexts and a CCD to record the recovered image. This article is arranged as follows. The encryption and decryption processes are presented in Section 2. The simulation results and discussions are given in Section 3. A brief conclusion is presented in Section 4.

2. Theoretical analysis of encryption process In our optical IBE system, the encryption process is performed digitally while the decryption process can be implemented optically or digitally. Let the function Oðu; vÞ represent the normalized intensity distribution of the original image, which is first modulated by a random POM Pðu; vÞ to constitute a complex-value image O0 ðu; vÞ as below pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ O0 ðu; vÞ ¼ Oðu; vÞPðu; vÞ ¼ Oðu; vÞexp½j2π randðu; vÞ where randðu; vÞdenotes a random function with uniform distribution in the range of [0,1]. The complex-valued image is then inverse Fresnel transformed digitally and decomposed into the following form: Dðξ; ηÞ ¼ FrTλ;d11 fO0 ðu; vÞg ¼ Eðξ; ηÞ þ Mðξ; ηÞ FrTλ;d11

ð2Þ

denotes the inverse Fresnel transform operator with where wavelength λ and propagation distance d1 . Mðξ; ηÞ ¼ exp½j2π r randðξ; ηÞ is a random POM, which is generated by computer in advance and has no relation with the inverse FrT spectrum Dðξ; ηÞ. Eðξ; ηÞ ¼ Dðξ; ηÞ  Mðξ; ηÞ is a complex field distribution. Wang's analysis has shown that the development of silhouette problem should be fully attributed to the phase distribution of Dðξ; ηÞ[27]. In the earlier IBE schemes using two POMs, the phase information of Dðξ; ηÞ is equally shared by both analytically obtained POMs. As a consequence, one can discern the silhouette information of the secret image using only one POM. Now, it is apparent that the phase information of Dðξ; ηÞ is not preserved in Eðξ; ηÞ due to the phase randomness of Mðξ; ηÞ. Thus, one cannot discern the visible information of the original image by just using Mðξ; ηÞ or Eðξ; ηÞ during the verification process. In the previous work, the complex

distribution Mðξ; ηÞ is regarded as the interference pattern of two analytically obtained POMs [28,29]. However, here, another approach is adopted to construct the encryption algorithm. Before proceeding with the later encryption procedures, we must present two properties of FrT that are used in the encryption-decryption method. Let the FrT of an image f ðxÞ at a propagation distance z when it is illuminated by a plane wave with wavelength λ be expressed as F z ðξÞ ¼ FrTλ;z ff ðxÞg. We have the following two properties for FrT [32]: FrTλ;z1 fFrTλ;z2 ½f ðxÞg ¼ FrTλ;z1 þ z2 ff ðxÞg and        j2π x0 x  jπ FrTλ;z exp f ðx  x0 Þ ¼ exp ð2ξx0 þ x20 Þ F z ðξÞ λz λz

ð3Þ

ð4Þ

where x0 is a real constant. The former property shows the additivity of propagation distances of FrTs. The latter demonstrates that the FrT has translation property that is very similar to the conventional Fourier transform (FT) when shifted input image is multiplied by a specially designed phase-only factor. It is straightforward to generalize those properties to the two-dimensional case. After generating the random POM Mðξ; ηÞ and complex-valued distribution Eðξ; ηÞ from the inverse FrT spectrum, two phase-only factors are employed to modulate these functions respectively in the following forms:   jπ E0 ðξ; ηÞ ¼ Eðξ; ηÞexp ð2ξx1 þ 2ηy1 þ x21 þ y21 Þ ð5aÞ λ d2    jπ  2ξx2 þ 2ηy2 þ x22 þ y22 ; M 0 ðξ; ηÞ ¼ Mðξ; ηÞexp λd2

ð5bÞ

where ðx1 ; y1 Þ and ðx2 ; y2 Þ are two sets of real constants that determine the center coordinates of ciphertexts in the input plane during the decryption processes and thus can be considered as additional keys of this proposal. Then, digital inverse FrTs are performed again to convert the modulated functions as below: C 1 ðx; yÞ ¼ A1 ðx; yÞexp½jθ1 ðx; yÞ ¼ FrTλ;d12 fE0 ðξ; ηÞg

ð6aÞ



C 2 ðx; yÞ ¼ A2 ðx; yÞexp½jθ2 ðx; yÞ ¼ FrTλ;d12 M 0 ðξ; ηÞ ;

ð6bÞ

where the obtained distributions C 1 ðx; yÞ and C 2 ðx; yÞ are final ciphertexts of the proposed scheme. The two ciphertexts can be assigned to different users for highly secure verification. For higher security level, the amplitude and phase components of the ciphertexts can also be provided to four different legal users. As for the translation coordinates ðx1 ; y1 Þ and ðx2 ; y2 Þ and FrT parameters, they can be transmitted to a principle decipher, who is in charge of the operation of decryption system. The silhouette of the original image cannot be discerned at the recovering plane if the potential decipher just acquires a single ciphertext. Moreover, the translation coordinates of ciphertexts in the input plane of decryption system are also significant for successful decryption. Fig. 1 illustrates the optoelectronic hybrid platform to complete decryption based on the optical interference principle. Besides a collimated coherent light source, the platform structure is very compact, in which only a complex SLM (or a combination of amplitude-modulated SLM and phase-modulated SLM) and a CCD are required. The two SLMs should work in transmission mode and need to be aligned strictly. Although single-beam structures have been proposed in some previous schemes by displaying the digital summation of all POMs (interference patterns) directly in the input SLMs [26–28], we choose alternative way to realize the single-beam decoding in this proposed scheme because more decoding key space will be introduced. Firstly, in the input complex SLM, the ciphertexts are shifted and modulated by two

Q. Wang et al. / Optics Communications 333 (2014) 151–158

the form of FrT at the distance d1 again. Finally, in the output plane, a CCD is utilized to record the decrypted image as below:

pure linear phase terms respectively as below:    j2π ðx1 x þ y1 yÞ C 1 ðx  x1 ; y y1 Þ Kðx; yÞ ¼ exp λd2    j2π ðx2 x þ y2 yÞ C 2 ðx  x2 ; y  y2 Þ þ exp λ d2

Oðu; vÞ ¼ jFrTλ;d1 fEðξ; ηÞ þ Mðξ; ηÞgj2 ¼ jFrTλ;d1 þ d2 fKðx; yÞgj2 ð7Þ

where the choice of ðx1 ; y1 Þ and ðx2 ; y2 Þ should avoid the overlapping of data. The pure linear phase terms and phase components of ciphertexts are displayed in the phase-only SLM, while the amplitude information are shown in the amplitude-only SLM. Then a collimated coherent beam with wavelength λ illuminates the joint input perpendicularly. From Eqs. (4)–(6), one can easily find that the following distribution will be achieved at the plane 1 through an FrT at the propagation distanced2 :  

 jπ FrTλ;d2 Kðx; yÞ ¼ E0 ðx; yÞexp ð2ξx1 þ2ηy1 þ x21 þ y21 Þ λd2    jπ 0 þ M ðx; yÞexp ð2ξx2 þ 2ηy2 þ x22 þ y22 Þ ¼ Eðξ; ηÞ þ Mðξ; ηÞ λd2 ð8Þ which is just the interference distribution of Eðξ; ηÞ and Mðξ; ηÞ. Next, the wave front distribution undergoes a free propagation in

K(x,y)

E (ξ,η ) + M (ξ,η )

O(u,v)

collimated beam λ SLMs input plane

153

ð9Þ

From the above analysis, one can find that the two sets of real constants given in Eq. (5) control the translation amount of ciphertexts and linear phase-only terms, which provide more protection to our system as additional keys. In the next section, numerical simulation results will be shown to verify the high sensitivity of decryption results to the translation coordinates. Using the proposed algorithm, one can simultaneously realize the single-beam decryption, the removal of silhouette problem and the enlargement of security key space. Here, it must be pointed out that so-called single-beam decryption structures of the previous IBE systems are actually not based on optical interference, but the digital summation of POMs [26–28]. Moreover, in the previous schemes, no more security key is introduced through single-beam decoding structure. In comparison with the linear phase blend operation of our previous work [28] and Kumar's jigsaw transform [26], the digital computational amount given in Eqs. (5) and (7) is simpler and more straightforward without any time-consuming iterative calculation involved. Although computational amount of Wang's scheme is also very simple [27], three SLMs must be used for optical implementation. In the following, we will give the computer simulations to show the feasibility and validity of our proposed method rather than optical demonstrations due to the resource limitation in our laboratory.

3. Numerical simulation and discussion

d2 plane 1 d1 (interference plane) Computer

Fig. 1. Optoelectronic implementation setup for decryption.

CCD

Numerical simulations have been conducted on the MATLAB 2009 R platform to verify the feasibility and security of this proposed scheme. Fig. 2(a) shows the primitive image used for test: a normalized Lena with the size of 128  128 pixels. The wavelength and propagation distances of FrTs are taken as λ ¼ 630 nm, d1 ¼100 mm and d2 ¼80 mm. Pixel size is set to be 16 mm. The random POM

Fig. 2. (a) The primitive images Lena; (b) the random POM Mðξ; ηÞ; (c) amplitude and (d) phase components of C 1 ðx; yÞ; (e) amplitude and (f) phase components of C 2 ðx; yÞ.

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Mðξ; ηÞ generated by computer in advance is shown in Fig. 2(b), which has no relation with the primitive image. Following the procedures given in Eqs. (1)–(6), one can obtain two complex-valued ciphertexts C 1 ðx; yÞand C 2 ðx; yÞ, the amplitude components and phase components of which are presented in Fig. 2(c)–(f). No information related with the original image can be recognized in each of them. During the encryption processes, the translation coordinates we choose arbitrarily for two ciphertexts are x1 ¼ 72, y1 ¼ 83, x2 ¼  54 and y2 ¼  105 (unit: pixel), respectively. As mentioned above, these coordinates and propagation parameters of FrT will serve as additional keys to protect the secret information. For correct decoding, in the input plane of optical verification setup, the ciphertexts must be shifted to the corresponding positions that are determined by aforementioned coordinates. In Fig. 3, relative positions of the ciphertexts are illustrated in the input plane of size 512  512 pixels, where the pixels outside of the ciphertexts are filled with zeros. Fig. 4(a) shows the decrypted image when correct ciphertexts assigned to different users are collected together and placed correctly. The relative error (RE) between the original image and the retrieved image is calculated to measure the quality of the retrieved image. The definition of RE is expressed as M

∑

N

∑ jjRðm; nÞj  jO0 ðm; nÞjj2

RE ¼ m ¼ 1 n ¼M1 ∑

ð10Þ

N

∑ jO0 ðm; nÞj2

m¼1n¼1

where M and N represent the image size, jRðm; nÞj and jO0 ðm; nÞj denote the amplitude distributions of the recovered image and the original image, respectively. The RE value calculated from Fig. 4 (a) is 1.2  10  0. Please note that this value is much larger than those obtained from some previous schemes (about 10–15–10–24) [23,28], which is mainly due to the clip of inverse FrT spectra during encryption processes. To generate the ciphertexts, we assume that the inverse FrT spectrum mainly distributes in a limited region (128  128 pixels). Spectrum distributions beyond this region are set to be zeros. In spite of this, the image quality shown in Fig. 4(a) is still acceptable without any discernible distortion or noise interference. Actually, in general, very small RE values (10–15–10–24) are not necessary for practical use.

Ciphertext 1

Ciphertext 2

Fig. 3. Relative positions of the ciphertexts in the input plane.

Then, we evaluate the effectiveness of this scheme to resolve the silhouette problem. In Fig. 4(b) and (c), we present the noiselike retrieved images by just using C 1 ðx; yÞ or C 2 ðx; yÞ for decryption, respectively, while other security keys are correctly employed. Using C 1 ðx; yÞ or C 2 ðx; yÞ, one can only obtain the distribution of Eðξ; ηÞ or Mðξ; ηÞ respectively in the interference plane (see Fig.1), neither of which contains the complete phase information of Dðξ; ηÞ that causes the silhouette problem. Thus, one cannot discern any remnant information of the primitive image in the retrieved results. The corresponding RE values are 0.652 and 0.641, respectively. To further analyze this issue, we investigated the influence of amplitude and phase components of ciphertexts on the decryption results. Fig. 4(d)–(g) show the recovered images when only one amplitude or phase component of ciphertext, that is, A1 ðx; yÞ, A2 ðx; yÞ, θ1 ðx; yÞ or θ2 ðx; yÞshown in Eq.(6), is unknown for illegal users (RE value: 0.604, 0.595, 0.564 and 0.573). In our tests, the unknown amplitude is assumed to be unity and the unknown phase is set to be zero. Although the corresponding RE values are not very large, above results have demonstrated that each amplitude or phase component is indispensable for successful recovering of the secret image. Now, we discuss sensitivity of this scheme to the translation coordinates of ciphertexts. Please note that in the following simulation tests, all the other keys remain correct when a wrong translation coordinate is employed in the decryption system. In Fig. 5(a)–(d), we present the decryption images obtained from correct ciphertexts that are shifted with wrong coordinate: x1 ¼ 71, y1 ¼ 82, x2 ¼  53 and y2 ¼  106, respectively. Although there is only one pixel distance (16 mm) between the wrong and correct coordinates in such tests, one cannot find any visible information related with the original image within these noise distributions (RE value: 0.667, 0.681, 0.674 and 0.659). To study the influence of translation coordinates analytically, we assume that incorrect coordinate x01 is used to control the position of ciphertext C 1 ðx; yÞ in the input plane of verification system while other coordinates and security keys remain correct. In this case, the expression of joint input K 0 ðx; yÞ can be easily obtained by just replacing x1 with x01 in the expression of K ðx; yÞ (Eq. (7)). Finally, through an FrT with distance d2 , one can get the distribution as below:  

 jπ 2 FrTλ;d2 K 0 ðx; yÞ ¼ E0 ðx; yÞexp ð2ξx01 þ 2ηy1 þ x02 þ y Þ 1 1 λd2    jπ ð2ξx2 þ 2ηy2 þ x22 þ y22 Þ þ M 0 ðx; yÞexp λd2   jπ ½2ξðx1  x01 Þ þ x21  x02 ¼ exp 1  Eðξ; ηÞ þ Mðξ; ηÞ λd2 ð11Þ In comparison with Eq.(8), it can be easily found that the noiselike result shown in Fig. 5(a) is mainly due to the linear phase-only term multiplied by Eðξ; ηÞ in Eq. (11). Above results imply that even though unauthorized user acquire all correct ciphertexts and propagation parameters of FrT accidentally, he/she cannot get any useful information of secret image yet without the knowledge of position coordinates of ciphertexts. Apparently, more security is achieved in this proposed scheme, compared with the earlier proposals that are purely based on optical interference. More importantly, to realize the enhancement of security level, we do not need to perform any time-consuming computation or to supplement more optical devices for decryption. Similar to the other encryption schemes in FrT domain, wavelength and propagation distance of FrT can be regarded as additional protection of this system. Note that here the correct propagation distance is d ¼ d1 þ d2 ¼ 180 mm. Fig. 6(a) and (b) display the recovered images obtained with wrong light

Q. Wang et al. / Optics Communications 333 (2014) 151–158

155

Fig. 4. (a) Retrieved image obtained with correct keys; (b) and (c) retrieved images produced from C 1 ðx; yÞ or C 2 ðx; yÞ; respectively; retrieved images obtained with the absence of (d) A1 ðx; yÞ, (e) A2 ðx; yÞ, (f) θ1 ðx; yÞ or (g) θ2 ðx; yÞ; respectively.

Fig. 5. Retrieved images when correct ciphertexts are shifted with wrong coordinates (a)x1 ¼ 71, (b)y1 ¼ 82, (c)x2 ¼  53 and (d)y2 ¼  106.

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Fig. 6. Retrieved images using (a) wrong wavelength and (b) wrong propagation distance.

0.8

0.8 0.7

0.6

0.6

RE

RE

0.5 0.4

0.4

0.3

0.2

0.2 0.1 0 −6

−4

−2

0

2

Deviation of wavelength

4

6 −4 x 10 nm

0 −4

−2 0 2 4 −7 Deviation of propagation distance x 10 m

Fig. 7. RE curves for (a) deviation of wavelength and (b) deviation of propagation distance of FrT.

C 0 ðx; yÞ ¼ Cðx; yÞð1 þ kQ Þ

ð12Þ

1 C C 0.8

1 2

0.6 RE

wavelength 630 þ4  10  4 nm or wrong propagation distance d ¼ 0:18 þ 3  10  7 m, respectively, in the verification system. In Fig. 7(a) and (b), the dependence of RE on the variations of wavelength and the propagation distance are presented in the form of curves within the neighborhood of correct values. One can find that all RE curves increase greatly, in a very narrow deviation range, around the required value. If RE ¼0.5 is selected as the threshold value, then the wavelength sensitivity is about 2  10  4 nm and distance sensitivity is about 1.5  10  7 m. To illustrate the sensitivity more clearly, decrypted images corresponding to the threshold value are also given as insets in the Fig. 7(a) and (b). One can still find some silhouette of the primitive images. As additional keys, these parameters of the FrT can further enlarge the key space of our system and improve resistance against brute force attacks. On the other hand, the high sensitivities of secret keys also pose stringent requirements for the adjustment precision of the implementation system, including the step width of laser wavelength, the variation error of propagation distance and alignment of two SLMs. Moreover, the influence of the SLM dimensions, numerical aperture, the size of pixel of the camera, and so forth on the quality of recovery also needs to be considered in optical experiment. If optoelectronic instruments at hand cannot meet the technique requirements, digital decryption is also a sensible choice. To evaluate the robustness of this proposed scheme, we designed a noise attack [22] to disturb the complex-valued ciphertexts as below

0.4

0.2

0

0

0.05

0.1 0.15 Noise intensity k

0.2

Fig. 8. The robustness test of noise attack.

where the function C 0 ðx; yÞ represents the ciphertext affected by noise, the coefficient k characterizes the noise intensity, and the function Q is a random Gaussian distribution with the mean value 0 and standard deviation 1. In the numerical simulation tests, one ciphertext is affected by noises with different intensities while another one remains unaffected. Fig. 8(a) shows the calculated RE values of decrypted image for different noisy intensities. In the

Q. Wang et al. / Optics Communications 333 (2014) 151–158

insets of Fig. 8, we present the decrypted images when C 1 ðx; yÞ or C 2 ðx; yÞ is influenced by noise with k ¼ 0:08, respectively, from which one can still find some visible information of the original image. Although certain robustness can be obtained for the proposed scheme, we must point out that the encryption schemes based on optical interference principle are, in general, more vulnerable to noise attacks due to the extremely high sensitivity, in comparison with many other proposals. As mentioned above, extremely high sensitivities to the wavelength and propagation distances of FrT pose serious technique problems for optical implementation of this proposal. A little frequency shift (10  3 nm) of laser wavelength, which cannot be

K(x,y)

E (ξ,η ) + M (ξ,η)

157

avoided in general, will lead to the failure of decryption. These problems can be alleviated by extending the proposed scheme to the Fourier transform (FT) domain. In this case, all of the FrT and inverse FrT operations required in this proposal are replaced by FTs and inverse FTs, respectively. Moreover, the modulating calculations of Eðξ; ηÞ and Mðξ; ηÞ shown in

0.7 0.6 0.5

O(u,v) RE

0.4 collimated beam

0.3

λ

SLMs input plane

0.2 f

f

f

f

plane 1 (interference plane)

CCD

Computer Fig. 9. Optoelectronic implementation setup used for decryption in the Fourier transform domain.

0.1 0 629.5

630 Wavelength of laser source

630.5 nm

Fig. 11. RE curve with respect to deviation of wavelength.

Fig. 10. (a) Retrieved image obtained with correct keys; (b) and (c) retrieved images produced from C 1 ðx; yÞ or C 2 ðx; yÞ; respectively; (d) retrieved image when correct ciphertexts are shifted with wrong coordinates x1 ¼ 71.

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Eq. (5) should be changed as   j2π ðξx1 þ ηy1 Þ E0 ðξ; ηÞ ¼ Eðξ; ηÞexp λf M 0 ðξ; ηÞ ¼ Mðξ; ηÞexp

  j2π ðξx2 þ ηy2 Þ λf

ð13aÞ

ð13bÞ

where f is the focal length of the transform lens. One can obtain   two by carrying out digital inverse FTs of E' ξ; η and  ciphertexts  M' ξ; η ; respectively. To extract the secret information, the ciphertexts are firstly shifted and Fourier transformed as  

 j2π ðξx1 þ ηy1 Þ FT C 1 ðx  x1 ; y  y1 Þ þ C 2 ðx  x2 ; y  y2 Þ ¼ E0 ðx; yÞexp λf þ M 0 ðx; yÞexp



  j2π ðξx2 þ ηy2 Þ ¼ Eðξ; ηÞ þ Mðξ; ηÞ λf

ð14Þ

By performing FT with transform lens again, the decrypted image is recorded by a CCD in the Fourier spectrum plane. In comparison with Eqs. (5) and (7), one can find that the computation complexity in the alternative scheme is reduced. However, two transform lenses must be introduced for FT operations at the optoelectronic implementation platform, as shown in Fig. 9. One can recover the secret image successfully (Fig. 10(a), RE ¼2.14  10  0) when the true ciphertexts are shifted to the correct positions respectively in the input SLMs and correct transform parameters such as wavelength are used. Fig. 10(b) and (c) display the noisy recovered images obtained only from the ciphertext C 1 ðx; yÞ or C 2 ðx; yÞ (RE: 0.617 and 0.634). The reconstructed image shown in Fig. 10(d) is obtained when ciphertext C 1 ðx; yÞ is shifted with wrong coordinates x1 ¼ 71 (RE ¼0.521). Finally, we study the dependence of decrypted results on the wavelength of laser source using RE as criterion. From the RE curve shown in Fig. 11, one can easily find that the sensitivity of the alternative encryption scheme based on FTs is about 0.5 nm if Re¼ 0.5 is still used as threshold value. Obviously, rigorous technique requirement for laser source is relaxed greatly compared with the aforementioned scheme in FrT domain, which is no doubt advantageous for optical implementation. On the other hand, the cost we must pay is the decline of security level as the sensitivity of the scheme to wavelength decreases. Moreover, optoelectronic implementation setup for information retrieval becomes complicated with two transform lens involved. 4. Conclusions In conclusion, we have proposed a new method for image encryption based on the optical interference principle and an analytical algorithm is proposed. Using this proposed algorithm, one can simultaneously realize the removal of silhouette problem, the single-beam implementation of decryption, and appreciable enhancement of security level. No time-consuming iterative computation is involved. Moreover, one can complete the decryption using compact setup without beam splitting. In this scheme, through a digital inverse FrT and random phase modulation, the

information of original image is firstly decomposed into a random POM and a complex distribution, in either of which phase component of inverse FrT spectrum, the main cause of the silhouette problem existing in the earlier IBE schemes, is not reserved completely. Then the obtained random POM and complex distribution are multiplied with two predefined linear phase-only terms, respectively. Performing digital inverse FrT again, one can achieve a pair of complex ciphertexts. To recover the secret image correctly, the ciphertexts must be shifted to the certain positions of the input plane of decryption system, respectively. The key space of this proposal is enlarged greatly as the translational coordinates is introduced as additional keys. The recovered images can be recorded in the CCD plane by carrying out an optical or digital FrT with correct propagation parameters on the joint input of two ciphertexts. Simulation results have shown the validity of the proposed approach. An alternative scheme in FT domain has also been discussed briefly to improve the application value of the proposed algorithm.

Acknowledgment This research is financially supported by the program of the National Natural Science Foundation of China (61101204 and 61107029). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

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