Efficient encrypting procedure using amplitude and phase as independent channels to display decoy objects John Fredy Barrera1,* and Roberto Torroba2 1
Grupo de Óptica y Fotónica, Instituto de Física, Universidad de Antioquia, A.A. 1226, Medellín, Colombia 2
Centro de Investigaciones Ópticas (CONICET-CIC) and UID OPTIMO, Facultad de Ingeniería, Universidad Nacional de La Plata, P.O. Box 124, La Plata (1900), Argentina *Corresponding author:
[email protected] Received 10 December 2008; revised 20 April 2009; accepted 7 May 2009; posted 13 May 2009 (Doc. ID 105156); published 1 June 2009
Objects acting as inputs of encrypting optical systems can be regarded as having two independent channels: amplitude and phase. In this context, we can use the term “complex objects” to refer these input objects. In this work we explore the way to perform an undercover operation where one channel (amplitude) is used to depict decoy information to confuse intruders, while the other (phase) operates with the true information. Besides, we use the Gerchberg–Saxton algorithm to transform the amplitude and phase encrypted information into pure phase data, therefore increasing the efficiency of the technique as only a single matrix containing these data needs to be sent. Finally, as an example to show the potential of the method, we combine the separate channels in a multiplexing technique with the Gerchberg– Saxton algorithm to generate an efficient multiuser secure process. © 2009 Optical Society of America OCIS codes: 100.4998, 070.4560, 100.0100, 070.0070.
1. Introduction
Optical encryption is a modern way to protect information from intruders [1–10]. Amplitude or alternatively phase of a given input object is processed by optical architectures using random phase masks as encoding keys. Input objects can be regarded as complex objects when simultaneously considering their amplitude and phase. The original work by Réfrégier and Javidi [1] first proposed a double random phase optical encryption system with two independent random phase masks on both the input plane and the Fourier plane that encode an input image as stationary white noise. The images introduced in the above original arrangement were amplitude representations. Later, the fully phase version was considered and compared with amplitude inputs [11]. In this last pa0003-6935/09/173121-09$15.00/0 © 2009 Optical Society of America
per, results revealed that the decrypted images obtained from fully phase encryption are more robust to additive noise than those obtained from amplitude-based encryption are. In another analysis [12] nonlinear (fully phase) encryption is probed to be more secure than linear (amplitude) encryption. If the original image is phase encoded and encrypted, then it is impossible to acquire the information content of the encrypted phase image unless we use a technique for converting the phase image into an amplitude image [13–15]. In another contribution [16] a method was introduced to encrypt multiple input objects using a single random phase mask and different amplitude masks in contact. During the whole procedure, the same phase mask is used for all objects, while a different amplitude mask is employed for each object. If during decryption we use the phase mask but no amplitude masks used during encryption are introduced, the superposition of all decrypted original objects is obtained. Therefore, amplitude masks, besides 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS
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making possible the multiplexing operation, act as selecting tools to choose the desired decrypted object. Likewise, Arizaga et al. [5] developed a simple optical architecture to encrypt the object amplitude data using an interference with a speckled reference wave, and the encrypted image is digitally stored by holography. In another approach, an optical double random phase encryption method with a joint transform correlator architecture that does not require an accurate optical alignment, was proposed [17,18]. The joint transform correlator architecture was also tested in a method for a fully phase-encoded key and the use of biometrics standards in security tasks. In this application, the phase masks are verified against a specific phase function that is stored in an optical pattern recognition system. Additionally, a fully phase encryption system using a fractional Fourier transform to encrypt and decrypt a 2-D phase image obtained from an amplitude image was implemented [19]. The encrypted image is holographically recorded in a barium titanate crystal and is then decrypted by generating through phase conjugation a conjugate of the encrypted image. The decrypted phase image is converted into an amplitude image by the phase contrast technique using an electrically addressed spatial light modulator. However, to our knowledge, there are no communications reporting the use of both encrypted amplitude and phase as separate channels in an undercover operation. We are primarily interested in the idea that the amplitude can be used as a decoy element to a hacker capturing the corresponding decoding mask sent through an open channel. We also remark that we need first to recover the amplitude information and then to perform the phase retrieval. Additionally, we analyze the case of transforming the completely amplitude and phase encrypted information into pure phase data. After this conversion, we need only send a single matrix containing these phase data and not send two separate matrices, one for the amplitude and the other for the phase information. Thus, we are increasing the efficiency of the technique besides keeping the robustness of phase information. We perform the phase transformation via the Gerchberg–Saxton algorithm [20,21]. The Gerchberg–Saxton algorithm was originally developed for applications such as superresolution, phase retrieval, and beam shaping. We also extend the technique to a case study of image multiplexing by placing in contact phase masks in the Fourier plane of a 4f encrypting architecture. A simple superposition of masks allows the encryption of different secret input complex images. Afterwards, the encrypted information is multiplexed on a single medium and then phase transformed using the Gerchberg–Saxton algorithm. An intruder not knowing the multiplexing operation but intercepting the phase encoding masks separately will reveal only the amplitude decoy images, 3122
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while every single authorized receiver will be instructed on the right superposition between phase masks to decode the right amplitude images. Finally, the authorized user has to employ the respective right phase decoding mask to recover the phase object. We present simulations that corroborate our proposal. Our main intention in this contribution is to show the expanding possibilities for the encrypting mechanism we have, when considering the separate channels for amplitude and phase, together with the higher efficiency achieved with the use of the Gerchberg–Saxton algorithm and the multiplexing procedure. 2.
Complex-Object Encoding
Let us remember the steps involved in an encrypting procedure using a classical 4f double phase mask architecture such as the one shown in Fig. 1. Let x and y denote the spatial coordinates and v and w denote the coordinates in the Fourier domain. Let COðx; yÞ ¼ f ðx; yÞ · exp½i2πqðx; yÞ denote the normalized complex information to be encrypted, where 0 < f ðx; yÞ < 1. Also, let pðx; yÞ and bðv; wÞ be two independent white sequences, uniformly distributed on the interval [0,1]. The optical field at the Fourier plane of the first lens is gðv; ωÞ ¼ FTv;w ff ðx; yÞ · exp½i2πqðx; yÞ · exp½i2πpðx; yÞg · exp½i2πbðv; wÞ;
ð1Þ
where FTf·g represents the Fourier transform operation. After the second Fourier transform, the encrypted image is represented by ψ a ðx; yÞ ¼ ff ð−x; −yÞ · exp½i2πqð−x; −yÞ · R1 ð−x; −yÞg ⊗ FTx;y fR2 ðv; wÞg;
ð2Þ
where ⊗ denotes the convolution operation, R1 ðx; yÞ ¼ exp½i2πpðx; yÞ is the input encoding mask, and R2 ðv; wÞ ¼ exp½i2π bðv; wÞ is the second phase encoding mask. Decryption is achieved by obtaining the complex conjugate of the encrypted image, followed by its Fourier transform, kðv; wÞ ¼ FTv;w ff ð−x; −yÞ · exp½i2πqð−x; −yÞ · R1 ð−x; −yÞg · R2 ðv; wÞ;
ð3Þ
where * denotes complex conjugate. After multiplying the optical field by phase mask R2 ðv; wÞ and a second Fourier transform, we get Dðx; yÞ ¼ f ðx; yÞ · exp½−i2πqðx; yÞ · R1 ðx; yÞ:
ð4Þ
At this stage, any intensity detector will record only jf ðx; yÞj2, thus eliminating all traces of phase information included in both object and first encoding mask.
Fig. 1. Double phase mask encrypting architecture: R1 and R2 , encrypting masks; L, transforming lens with focal length f ; * denotes complex conjugate. The upper and lower parts describe the complex encrypyting and decrypting procedures respectively.
Since f ðx; yÞ is positive by definition, we may obtain the amplitude information by taking the magnitude of jf ðx; yÞj. This method is referred to in the literature as the intensity decryption method. If we intend to recover the fully complex object (amplitude and phase), we additionally need to multiply by the first encoding mask R1 ðx; yÞ. This operation allows us to retrieve the complex conjugate of the original complex object, CO ðx; yÞ ¼ f ðx; yÞ · exp½−i2πqðx; yÞ:
ð5Þ
We include in Fig. 2 a block diagram to illustrate the process. Computational simulations performed on a Matlab platform with 4096 × 4096 matrices and 8 bit resolution present the potential of the method. The illuminating wavelength was 632:8 nm. As a way of illustration, we insert in the 4f architecture a complex input object, depicting in Fig. 3(a) its binary amplitude and in Fig. 3(b) its phase. As usual, an input-plane phase mask also multiplies this complex object. As described above, a second phase mask at the Fourier plane encodes the information, and a CCD camera captures these data. We first reconstruct, using the Fourier plane phase mask, the amplitude component of our input object as shown in Fig. 3(c) with no traces of the phase component. When also using the input-plane phase mask and, for instance, a phase contrast technique, the phase component may be recovered. Figure 3(d) shows a reconstruction using our digital processor.
Note that the phase information of the decrypted image [Fig. 3(d)] corresponds to the complex conjugate of the original information [Fig. 3(b)], as evident from Eq. (5). Regarding our proposal, we can assume that this last code would be the actual authentication to some operation allowed to the authorized user. The major concept in handling the input object as a complex object is the availability of separate channels, which can be independently exploited. There are several advantages to mention in the concept of encrypting complex objects: a. We have two independent channels for the encrypted data (amplitude + phase).
Fig. 2. Block diagram for (a) encryption and (b) decryption. FT represents the Fourier transform operation, and * denotes complex conjugate. 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS
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Fig. 3. (a) and (b) Binary amplitude and phase, respectively, of the complex input object (note the message in the bars in the phase information); (c) reconstructed amplitude using only the Fourier-plane coding mask, observing no traces of the phase component; (d) phase component reconstructed using the input-plane phase mask.
b. We can use the concept of hierarchies, as a publicly delivered key (Fourier plane phase mask) allows recovering the amplitude information, while a privileged key (input-plane phase key) grants restricted access to the phase information. In this case, both amplitude and phase are valid information. c. We can recover partial information in a time sequence, with the same encrypted complex object. Authorized users can recover the amplitude information in a certain instance, which would be useful at that moment. In a later opportunity, without sending another encrypted object, but now receiving the input-plane-encrypting mask, the phase information can be read. d. We can use the amplitude information as a decoy to intruders who accessed to the Fourier plane phase masks, while phase information (true information) requires an additional procedure (input-plane phase mask).
Fig. 4 shows the applied Gerchberg–Saxton algorithm. In a clockwise sense, the Fourier transform and the inverse Fourier transform are used in steps 1 and 3, while constraints of output and input beams are realized in steps 2 and 4, respectively. During the
3. Applying the Gerchberg–Saxton Algorithm
Once we have achieved the encrypted data, the next step is to find a more efficient way to send these complex data to the prospective authorized users. At this point, we introduce the Gerchberg–Saxton algorithm in the proposed procedure. The block diagram in 3124
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Fig. 4. Block diagram showing the sequence (clockwise) when applying the Gerchberg–Saxton algorithm. FT and FT−1 represent the direct and inverse Fourier transform operation, respectively.
last step, we use the mean square error to compare the initial and final amplitudes E1 and E1 0 . If a value of the mean square error does not belong to the assumed range, we force equalization ϕ1 ¼ ϕ0 1 and the iteration loop begins once again. When the mean square error is within the assumed range the iteration stops getting the information of the final phase ϕ0 1 . The main advantage in using the Gerchberg– Saxton algorithm is that instead of sending matrices with phase and amplitude information to the users, we send only a phase matrix, therefore making the whole process more efficient. After applying the Gerchberg–Saxton algorithm to the final encrypted images, we got a phase-only element. We will show that amplitude and phase information are both completely recovered from the pure phase encrypted image. We intended this operation for multiple end users. Each one privately gets the corresponding input-plane phase mask and the corresponding Fourier-plane phase masks. We encrypt the complex object depicted in Figs. 5(a) and 5(b). Thereafter, we convert the resulting encoded image into a pure phase image [Fig. 5(c)]. We obtain, during decryption, the phase element complex conjugate, and subsequently we perform a Fourier transform. We multiply by the corresponding mask to decrypt the information of the complex object amplitude. Finally, we have to multiply by the input encoding mask to recover the input objects phase component. In Figs. 5(d) and 5(e), we exhibit the reconstructed amplitude and phase components of the complex objects, respectively. Evidently, this approach is possible thanks to the discussion developed in the preceding sections. 4. Example of Application to Multiplexing Encryption
After introducing the idea of complex object encryption and the advantages of the Gerchberg–Saxton algorithm to make the encrypted image sending more efficient, we want to show the potential of this combination by giving a practical application. We choose to implement a multiplexing of encrypted data by proposing a modification to the multiple phase key approaches in 4f architectures. The use of the independent amplitude and phase channels allow us to introduce decoy object in one of these channels to confuse the eventual hackers. We decided on multiplexing for two main reasons: 1. Several encrypted objects can be sent in a single image package. Various users may get the encrypted information and according to their authorized keys, and they recover the pertinent information. 2. In the event of intrusion, the existence of multiple keys leads the intruder to recover multiple data not knowing which one may be useful. We find in the literature methods that use several random phase masks to increase the security of the
Fig. 5. Image of an input complex object to be encrypted and phase transformed with the aid of the Gerchberg–Saxton algorithm. (a) and (b) Amplitude and phase of the object, respectively. (c) Pure phase element obtained after applying the Gerchberg– Saxton algorithm over the complex encrypted object. The resulting decrypted image corresponding to the input complex object is shown in (d) amplitude and (e) phase.
optical encrypting systems [22–25]. A system based in cascade phase keys to achieve a high security level has been described [22]. Meng et al. [23] presented a versatile security method that employs a series of phase masks cascaded in free space with a higher degree of freedom in key space. The sandwich between two diffusers in the Fourier plane in a double random phase mask encoding method was developed to increase the reliability of the process [24]. Additionally, it is possible to perform a secure multiplexing procedure using the sandwich of two diffusers whether during the encryption and decryption of each object one of the diffusers rotates a specific amount [25]. Following these proposals, we present another alternative. Our idea is to introduce a multiplexing 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS
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encryption using, as security keys, the product of random phase masks in the Fourier plane in the 4f optical architecture. We want to combine this new alternative together with the two channels brought by considering amplitude and phase as independent means. In our proposal, one of the masks at the Fourier plane is called a “permanent mask” and the others are called “changing masks” as they are changed every time a new input object is processed. The flow chart in Fig. 6 shows the encrypting multiplexing procedure when using three inputs. Complex input objects (CO1 , CO2 , and CO3 ) are bonded to inputplane phase mask R1 . The permanent mask in the Fourier plane is denoted by R2. Encoding of CO1 is performed with R2 and R3 in contact, encoding of CO2 is performed with R2 and R4 in contact, while encoding of CO3 is done with R2 , R3 , and R4 in contact. The encrypted information E1 , E2 , and E3 is then added to build M. The decryption step by using M allows the recovery of the complex conjugate of
the original complex objects CO1 , CO2 , and CO3 when the appropriate combinations R2 R3, R2 R4 , or R2 R3 R4 are used together with R1 . According to Eq. (5), multiplexed encrypted images are represented as Mðx; yÞ ¼ ½CO1 ð−x; −yÞ · R1 ð−x; −yÞ ⊗ FTx;y fR2 ðv; wÞ · R3 ðv; wÞg þ ½CO2 ð−x; −yÞ · R1 ð−x; −yÞ ⊗ FTx;y fR2 ðv; wÞ · R4 ðv; wÞg þ ½CO3 ð−x; −yÞ · R1 ð−x; −yÞ ⊗ FTx;y fR2 ðv; wÞ⋅R3 ðv; wÞ · R4 ðv; wÞg:
ð6Þ
The multiplexed decrypted field, after a complex conjugate operation and a Fourier transform, has the form km ðv; wÞ ¼ ½FTv;w fCO1 ð−x; −yÞ · R1 ð−x; −yÞg · R2 ðv; wÞ · R3 ðv; wÞ þ ½FTv;w fCO2 ð−x; −yÞ · R1 ð−x; −yÞg · R2 ðv; wÞ · R4 ðv; wÞ þ ½FTv;w fCO3 ð−x; −yÞ · R1 ð−x; −yÞg · R2 ðv; wÞ · R3 ðv; wÞ⋅R4 ðv; wÞ:
ð7Þ
As an example, if we wish to recover the object CO1 , we have to multiply at the Fourier plane by random masks R2 ðx; yÞ and R3 ðx; yÞ, and then, after a second Fourier transform we have D1 ðx; yÞ ¼ CO1 ðx; yÞ · R1 ðx; yÞ þ ½CO2 ðx; yÞ · R1 ðx; yÞ ⊗ FTx;y fR4 ðv; wÞ · R3 ðv; wÞg þ ½CO3 ðx; yÞ · R1 ðx; yÞ ⊗ FTx;y fR4 ðv; wÞg:
Fig. 6. Flow charts depicting the multiplexing procedure for a three-input case: CO1 , CO2 , and CO3 , complex input objects; I , Fourier transform; R1 , R2 , R3 , and R4 , random phase masks; E1 , E2 , and E3 encrypted complex objects; M, multiplexed encrypted information; *, complex conjugate. 3126
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ð8Þ
The first term in Eq. (8) shows the complex conjugate optical field at the input plane during the first input object encryption. We see the first object complex conjugate times the first random mask from the input plane. This term allows obtaining the amplitude and phase from the first complex object. In turn, the second and third terms in Eq. (8) contribute as background noise, because the remaining images remain encrypted. Note that the noise over the decrypted image increases as the number of encrypted images increases. In the next paragraph, we briefly describe the practical way our security system works. We send separately on a public channel the Fourier-plane pure phase masks. On a separate private channel, as usual, the authorized user gets the right encrypting input-plane phase masks along with the information on the superposition of all the Fourier-plane masks to recover the third object. We consider this last object
Fig. 7. Images of two input complex objects to be encrypted, multiplexed, and phase transformed with the aid of the Gerchberg– Saxton algorithm. (a) and (b) Amplitude and phase of one object, respectively; (c) and (d) second object amplitude and phase, respectively.
the valid information. Without the appropriate knowledge about the superposition of all Fourier plane phase masks and the appropriate input-plane phase mask, a hacker will eventually recover the two first objects, which act as decoy samples. We encrypt the two complex objects depicted in Fig. 7 according to the protocol described in the present section . In addition, after multiplexing operation we convert the resulting encoded image into a pure phase image using the Gerchberg–Saxton algorithm. We obtain, during decryption, the phase element complex conjugate, and thereafter we perform a Fourier transform. We multiply in each case by the corresponding mask (resulting from the product of the permanent mask and the adequate changing mask) to decrypt the complex object amplitude information. Finally, we have to multiply by the input encoding mask to recover the input objects’ phase component. In Fig. 8, we exhibit the reconstructed amplitude and phase components of both input complex objects. Evidently, this approach is possible thanks to the discussion developed in the preceding sections. It is important to note that the Gerchberg–Saxton algorithm and the multiplexing technique generate an efficient multiuser process that allows expanding the capacity of the double random phase encryption. In order to analyze the effect of contaminating noise on the multiplexed encrypted images transformed into pure phase information, we simulated the noise by randomly converting a portion of the pixels in the image into black pixels. This represents the equivalent to data loss. We use the normalized mean square error (NMSE) metric to construct the plot depicted in Fig. 9. This metric follows the expression
Fig. 8. Resulting decrypted images corresponding to the objects shown in Fig. 7. The reconstructions depict in (a) and (c) the objects amplitudes, and in (b) and (d) their respective phases.
NMSE ¼
N 1 X · jDðm; nÞ − D0 ðm; nÞj2 ; K m;n
ð9Þ
where ðm; nÞ are the pixel coordinates, N × N is the number of pixels of the image, Dðm; nÞ is the decrypted object obtained without introducing noise over the pure phase multiplexing, D0 ðm; nÞ is the decrypted information when there are different percentages of data loss, and K is the MSE between Dðm; nÞ and the worst expected case Dw ðm; nÞ. K¼
N X
jDðm; nÞ − Dw ðm; nÞj2 :
ð10Þ
m;n
The multiplexed encrypted image is openly sent to the user, and it could be affected by noise. Therefore,
Fig. 9. NMSE curve as a fuction of the percentage of random noise affecting the pure phase multiplexing. 10 June 2009 / Vol. 48, No. 17 / APPLIED OPTICS
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this image is the most vulnerable candidate to the external noise pollution. If a recognizable image in the decryption process is considered the limit of the acceptable data loss, a NMSE ¼ 0:8 will be the value of maximum tolerance. This corresponds to a random noise of around 50% as can be seen in Fig. 9. When the data loss area exceeds the 50%, major relevant image information is missed. This analysis reveals the robustness of the proposed protocol, regarding the influence of data loss. We have to point out that each image is encrypted with statistically independent masks; therefore nondecrypted images remain as background white noise over the decoded image, thus avoiding the existence of potential cross-talk [see Eq. (8)]. In our application, the efficiency of the technique takes priority over the speed of the computational resources. The major objectives are two: first, to handle multiple encrypted data in only one sending, second to deal with complex objects (both amplitude and phase). Third, we introduce an additional advantage, which is to reduce the amount of data transferred to the end user by sending a single matrix containing only phase information. Here is where the Gerchberg–Saxton algorithm plays a role as an instrument to reduce the information to be sent by a factor of two. The combination of these three purposes makes, according to our point of view, the whole process more “efficient.”
phase information, thus reducing the amount of information to be sent. Furthermore, in the example we introduce, the multiplexing procedure increases the amount of data to be processed and improves the security of the protocol. The combination of the Gerchberg–Saxton algorithm and the multiplexing technique permits performing an efficient and robust multiple encryption–decryption procedure. This last is an indication of the potential implicit in our contribution, allowing expansion to an upper limit of the possibilities of the 4f encrypting scheme. We are aware that due to the linearity of the optical procedures, the 4f setup is vulnerable to different attacks. The proposed protocol offers an alternative to enhance the security level, without pretending to say that the system is immune to those attacks. This research was performed under grants from the following organizations: Instituto Colombiano para el Desarrollo de la Ciencia y la Tecnología (COLCIENCIAS, Colombia), Comité para el Desarrollo de la Investigación (CODI)–Universidad de Antioquia (Colombia), Consejo Nacional de Investigaciones Cientifícas y Técnicas (CONICET) No. 5995 (Argentina), Academia Nacional de Ciencia y TecnologíaPromoción Cientifíca y Tecnología (ANCYT PICT 12564, Argentina), and Facultad de Ingeniería, Universidad Nacional de La Plata No. 11/I105 (Argentina).
5. Conclusions
References
We present an alternative for operating in encryption, by using the full capacity of both amplitude and phase of a given object as separate channels. Under this perspective, we can work with two independent channels, not necessarily complementary to each other for decrypting purposes. Different information can be coded in amplitude and phase, in such a way that an amplitude-authorized user can receive the phase encoding but without the appropriate key is unable to read it. This situation generates the possibility of hierarchical users, associated with the idea of public and private decoding keys. For example, some users receive the amplitude key (public key), allowing them to acquire the object amplitude, while only a restricted number get the first encoding key (private key) besides the amplitude key, permitting them to access to the object phase information. In a further improvement, the authorized users, without knowing, have access to encoded information that can be released in time, as they can get the phase key in the a posteriori instance. We can also think that amplitude information is a just decoy for unwanted intruders accessing to the encrypted information and the Fourier plane encoding phase mask. Assuming the true information is that contained in the phase part, the intruder needs the first encoding phase mask to obtain the complete image, but ignoring this fact he/she fails in cracking the system. Additionally, we use the Gerchberg–Saxton algorithm to convert the encrypted channels into pure
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